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An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

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Abstract

A vertex centred Jameson–Schmidt–Turkel (JST) finite volume algorithm was recently introduced by the authors (Aguirre et al., 2014 [1]) in the context of fast solid isothermal dynamics. The spatial discretisation scheme was constructed upon a Lagrangian two-field mixed (linear momentum and the deformation gradient) formulation presented as a system of conservation laws [2–4]. In this paper, the formulation is further enhanced by introducing a novel upwind vertex centred finite volume algorithm with three key novelties. First, a conservation law for the volume map is incorporated into the existing two-field system to extend the range of applications towards the incompressibility limit (Gil et al., 2014 [5]). Second, the use of a linearised Riemann solver and reconstruction limiters is derived for the stabilisation of the scheme together with an efficient edge-based implementation. Third, the treatment of thermo-mechanical processes through a Mie– Grüneisen equation of state is incorporated in the proposed formulation. For completeness, the study of the eigenvalue structure of the resulting system of conservation laws is carried out to demonstrate hyperbolicity and obtain the correct time step bounds for non-isothermal processes. A series of numerical examples are presented in order to assess the robustness of the proposed methodology. The overall scheme shows excellent behaviour in shock and bending dominated nearly incompressible scenarios without spurious pressure oscillations, yielding second order of convergence for both velocities and stresses.

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... Additionally, an ad-hoc strain smoothing procedure, typically employed in the context of Reproducing Kernel Particle Method, is required to avoid locking difficulties and instability issues. An alternative widely employed in the context of solid dynamics is the mixed-based methodology [36][37][38][39][40][41][42][43][44][45][46]. In this methodology, the motion of a deformable body is described using a system of first-order conservation laws. ...
... This substitution allows us to deduce ∇ 0 α. Once ∇ 0 α is determined, the shape function derivatives can finally be obtained from Eq. (41). Note that the method for computing derivatives described in (41), which is based on the complete linear polynomial basis, differs from the approaches typically used in the corrected SPH method [20,77,78]. ...
... Once ∇ 0 α is determined, the shape function derivatives can finally be obtained from Eq. (41). Note that the method for computing derivatives described in (41), which is based on the complete linear polynomial basis, differs from the approaches typically used in the corrected SPH method [20,77,78]. In the corrected SPH method, the gradient computation is explicitly modified to ensure first-order completeness. ...
Article
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This paper presents a new stabilised Element-Free Galerkin (EFG) method tailored for large strain transient solid dynamics. The method employs a mixed formulation that combines the Total Lagrangian conservation laws for linear momentum with an additional set of geometric strain measures. The main aim of this paper is to adapt the well-established Streamline Upwind Petrov–Galerkin (SUPG) stabilisation methodology to the context of EFG, presenting three key contributions. Firstly, a variational consistent EFG computational framework is introduced, emphasising behaviours associated with nearly incompressible materials. Secondly, the suppression of non-physical numerical artefacts, such as zero-energy modes and locking, through a well-established stabilisation procedure. Thirdly, the stability of the SUPG formulation is demonstrated using the time rate of Hamiltonian of the system, ensuring non-negative entropy production throughout the entire simulation. To assess the stability, robustness and performance of the proposed algorithm, several benchmark examples in the context of isothermal hyperelasticity and large strain plasticity are examined. Results show that the proposed algorithm effectively addresses spurious modes, including hour-glassing and spurious pressure fluctuations commonly observed in classical displacement-based EFG frameworks.
... It allows to account for different types of thermo-mechanical coupling at the constitutive level, like thermal softening and an equation of state, while the solution in the discrete setting of these conservations laws will ensure that the right shock speeds will be computed. Especially, this variational constitutive update is compatible with any numerical scheme dedicated to the approximation of the solution of hyperbolic systems, like finite volumes [5,22,34,45,1], finite elements [9], Discontinuous Galerkin methods [15,10], particle methods [28], eventually coupled with a discontinuous approximation [61,63,62,39]. Its particularity lies in that it is driven by updated values of some strain measure and of the internal energy density, hence getting rid of the two-field thermal formulation employed in [12,13,4,74], while the concept of pseudo-stresses introduced by Mosler and co-workers [49,50,7] is adopted. ...
... where F −T gathers columnwise the cobasis vectors e (α) , 1 ≤ α ≤ 3, such that F −T = (e (1) , e (2) , e (3) ). The cobasis vectors e (α) are defined as ...
... where (F e ) −T gathers columnwise the vectors of the elastic cobasisẽ (α) , 1 ≤ α ≤ 3, such that (F e ) −T = (ẽ (1) ,ẽ (2) ,ẽ (3) ). Equation (54) has a source term involving the plastic flow defined by Equation (27). ...
Preprint
In this paper, two Eulerian and Lagrangian variational formulations of non-linear kinematic hardening are derived in the context of finite thermoplasticity. These are based on the thermo-mechanical variational framework introduced by Heuzé et al. [37], and follow the concept of pseudo-stresses introduced by Mosler [48]. These formulations are derived from a thermodynamical framework and are based on the multiplicative split of the deformation gradient in the context of hyperelasticity. Both Lagrangian and Eulerian formulations are derived in a consistent manner via some transport associated with the mapping, and use quantities consistent with those updated by the set of conservation or balance laws written in these two cases. These Eulerian and Lagrangian formulations aims at investigating the importance of non-linear kinematic hardening for bodies submitted to cyclic impacts in dynamics, where Bauschinger and/or ratchetting effects are expected to occur. Continuous variational formulations of the local constitutive problems as well as discrete variational constitutive updates are derived in the Eulerian and Lagrangian settings. The discrete updates are coupled with the second order accurate flux difference splitting finite volume method, which permits to solve the sets of conservation laws. A set of test cases allow to show on the one hand the good behaviour of variational constitutive updates, and on the other hand the good consistency of Lagrangian and Eulerian numerical simulations.
... To assess the accuracy of δ-ULSPH-R-TIC, the same simulation of plate rotation is carried out by a well-developed and rigorously validated FVM code [68,69]. The FVM simulation is performed by using a second order vertex-centred finite volume algorithm combined with an explicit Runge-Kutta time integrator as well as using an acoustic ...
... Riemann solver together with linear reconstruction for the evaluation of the numerical fluxes (Aguirre et al. [68]; Hassan et al. [69]). In the FVM simulation, the plate was discretised by linear triangular elements with a spatial resolution of 25×25×2 (676 nodes/1250 elements). ...
... For further validation of δ-ULSPH-R-TIC, the simulation of elastic rings impact is repeated by using an FVM code [68,69], which is proven to be reliably applicable to structural mechanics problems including those characterised by material impact [71]. ...
Article
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This paper presents a set of novel refined schemes to enhance the accuracy and stability of the updated Lagrangian SPH (ULSPH) for structural modelling. The original ULSPH structure model was first proposed by Gray et al. (Comput Methods Appl Mech Eng 190:6641–6662, 2001) and has been utilised for a wide range of structural analyses including metal, soil, rubber, ice, etc., although the model often faces several drawbacks including unphysical numerical damping, high-frequency noise in reproduced stress fields, presence of several artificial terms requiring ad hoc tunings and numerical instability in the presence of tensile stresses. In these regards, this study presents a set of enhanced schemes corresponding to (1) consistency correction on discretisation schemes for differential operators, (2) a numerical diffusive term incorporated in the continuity or the density rate equation, (3) tuning-free stabilising term based on Riemann solution and (4) careful control/switch of stress divergence differential operator model under tensile stresses. Qualitative/quantitative validations are conducted through several well-known benchmark tests.
... Aiming to resolve the shortcomings described above, the main goal of this paper is to explore the solution of contact dynamics utilising a set of first order conservation equations [17][18][19][20], combined with the associated jump conditions across moving shocks 2 [21][22][23][24][25][26][27][28][29][30][31][32][33]. Building upon previous work developed by the authors [34,35], a mixed methodology is presented in the form of a system of hyperbolic conservation laws, where the linear momentum and the minors of deformation (the deformation gradient tensor, its co-factor and its Jacobian) are regarded as the main conservation variables of this mixed approach. Taking advantage of this formalism, appropriate kinetic and kinematic contact interface conditions can be suitably enforced at the boundary fluxes of the underlying hyperbolic system by means of the Rankine-Hugoniot jump conditions. ...
... Consider the three dimensional deformation of an isothermal body of material density ρ R moving from its initial undeformed configuration Ω V , with boundary ∂Ω V defined by an outward unit normal N , to a current deformed configuration Ω v (t) at time t, with boundary ∂Ω v (t) defined by an outward unit normal n. The time dependent motion ϕ(X, t) of the body can be described by the following system of Total Lagrangian global conservation laws [35,[39][40][41][42][43][44][45][46][47][48][49][50][51][52] ...
... This key concept will be further exploited in Section 5.2 at a semi-discrete level. The vertex centred finite volume spatial discretisation presented in this work requires the generation of a median dual mesh [34,35,50] for the definition of control volumes (see Figure 3). With this in mind, expression (1) can now be spatially discretised over an undeformed control volume Ω a V , to give ...
Article
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This paper presents a vertex‐centred finite volume algorithm for the explicit dynamic analysis of large strain contact problems. The methodology exploits the use of a system of first order conservation equations written in terms of the linear momentum and a triplet of geometric deformation measures (comprising the deformation gradient tensor, its co‐factor and its Jacobian) together with their associated jump conditions. The latter can be used to derive several dynamic contact models ensuring the preservation of hyperbolic characteristic structure across solution discontinuities at the contact interface, a clear advantage over the standard quasi‐static contact models where the influence of inertial effects at the contact interface is completely neglected. Taking advantage of the conservative nature of the formalism, both kinetic (traction) and kinematic (velocity) contact interface conditions are explicitly enforced at the fluxes through the use of appropriate jump conditions. Specifically, the kinetic condition is enforced in the usual linear momentum equation, whereas the kinematic condition can now be easily enforced in the geometric conservation equations without requiring a computationally demanding iterative algorithm. Additionally, a Total Variation Diminishing shock capturing technique can be suitably incorporated in order to improve dramatically the performance of the algorithm at the vicinity of shocks. Moreover, and to guarantee stability from the spatial discretisation standpoint, global entropy production is demonstrated through the satisfaction of semi‐discrete version of the classical Coleman–Noll procedure expressed in terms of the time rate of the so‐called Hamiltonian energy of the system. Finally, a series of numerical examples is examined in order to assess the performance and applicability of the algorithm suitably implemented in OpenFOAM. The knowledge of the potential contact loci between contact interfaces is assumed to be known a priori.
... Some researchers have tried to overcome these difficulties by enforcing incompressibility condition dictated by the space conservation law using FV discretization method. For example, it has been suggested that the conservation property of the FV method is used to solve linear momentum equation alongside with additional equations given by various geometric conservation laws in order to remove the undesirable features associated with the solution of nearly incompressible solids [21][22][23][24][25]. Alternatively, other researchers have introduced hydrostatic pressure as an unknown variable which must be sought in a way that ensures satisfaction of the incompressibility condition. ...
... So, keeping the same BC for the velocity, new boundary condition is proposed for the pressure which enforces the continuity as below: (22) [ ] Substituting ρ from Eq. (17) and expanding divergence term, results in the following expression: (23) For the velocity BC, the balance of the traction can be written as below: (24) Where t and P ext represent traction vector and external pressure applied at the boundary. Adopting the procedure suggested in [32], the normal gradient is obtained from traction balance as follows: (25) [ ] where τ is a time constant parameter. It should be noted that σ is the Cauchy stress which is a function of B and B is a function of u. ...
... More precisely, combining Eqs. (23) and (25) with τ=Δt and also using Euler scheme, pressure at these boundaries is calculated as follows: ...
Article
Unification of the numerical models and methods in computational solid and fluid dynamics has been a research objective with at least two major advantages in mind. The first benefit of such unification is the more efficient data transfer between fluid and solid media, and the second advantage is the possibility of developing a better solver as compared to the separate existing solid and fluid solvers. In this paper, a conservative fluid-like pressure-velocity-based formulation is proposed that simulates large deformation of a weakly compressible hyper-elastic solid on an Arbitrary Lagrangian-Eulerian (ALE) framework. The proposed solver, which is implemented in OpenFOAM software, allows for flexible grid movement, i.e. mesh points are not forced to follow material points, as well as for the employment of various material models such as Mooney-Rivlin and Neo-Hookean constitutive laws. Three challenging 2-D and 3-D test cases including torsion, bending and pressing of solid objects are presented to examine and discuss the accuracy and flexibility of the proposed solver. Furthermore, more light is shed on the concept of the pressure in compressible solids and appropriate boundary conditions at traction boundaries.
... By using commercial software, it is possible to numerically simulate a wide range of applications, such as fracture and fragmentation, metal forming and additive manufacturing processes, contact and hypervelocity impact [16]. These numerical tools are typically developed on the basis of classical low order finite element displacement-based formulations and, consequently, suffer from a certain number of numerical difficulties (locking, hour-glass modes and spurious pressure oscillations) [7,17,18]. Although some modifications have been applied to commercial tools to alleviate some of these shortcomings [14,[19][20][21], numerical difficulties are still evident when dealing with nearly/truly incompressible materials [16,22]. ...
... Although some modifications have been applied to commercial tools to alleviate some of these shortcomings [14,[19][20][21], numerical difficulties are still evident when dealing with nearly/truly incompressible materials [16,22]. Additionally, the shock-capturing technologies are poorly developed in the context of solid dynamics [6,7,18,23]. It is worthwhile noting that the development and extension of the current capabilities of these tools are not straightforward due to the closed nature of their implementation. ...
... In recent years, the research group at Swansea University have pursued the same {p, F } system whilst exploiting a wide range of spatial discretisation techniques including upwind cell centred FVM [6,18], Jameson-Schmidt-Turkel vertex centred FVM [118], upwind VCFVM [7,16], two step Taylor-Galerkin FEM [119] and stabilised Petrov-Galerkin FEM [17]. In subsequent papers, the {p, F } system was then augmented by incorporating a new conservation law for the Jacobian of the deformation J [7,16,23] to effectively solve nearly incompressible deformations. ...
Thesis
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It is evidently not trivial to analytically solve practical engineering problems due to their inherent nonlinearities. Moreover, experimental testing can be extremely costly and time-consuming. In the past few decades, therefore, numerical techniques have been progressively developed and utilised in order to investigate complex engineering applications through computer simulations. In the context of fast thermo-elastodynamics, modern commercial packages are typically developed on the basis of second order displacement-based finite element formulations and, unfortunately, that introduces a series of numerical shortcomings (e.g. detrimental locking, hour-glass modes, spurious pressure oscillations). To rectify these drawbacks, a mixed-based set of first order hyperbolic conservation laws for thermo- elastodynamics is presented in terms of the linear momentum per unit undeformed volume, the deformation gradient, its co-factor, its Jacobian and the balance of total Energy. Interestingly, the conservation formulation framework allows exploiting available CFD techniques in the context of solid dynamics. From a computational standpoint, two distinct spatial discretisations are employed, namely, Vertex-Centred Finite Volume Method (VCFVM) and Smooth Particle Hydrodynamics (SPH). A linear reconstruction procedure together with a slope limiter is employed in order to ensure second order accuracy in space whilst avoiding numerical oscillations in the vicinity of sharp gradients. The semi-discrete system of equations is then temporally discretised using a second-order Total Variation Diminishing (TVD) Runge-Kutta time integrator. Finally, a wide spectrum of challenging examples is presented in order to assess both the performance and applicability of the proposed schemes. The new formulation is proven to be very efficient in nearly incompressible thermoelasticity in comparison with classical finite element displacement-based approaches.
... Consideration of thermal effects, especially in the context of large strain fast transient dynamics, is fundamental in order to obtain a realistic representation of stresses when a solid undergoes a complex and rapidly evolving thermally-coupled deformation pattern [33][34][35][36][37][38][39][40][41]. With focus on thermo-elasticity and thermo-inelasticity, numerous authors have worked over the years putting forward a variety of computational schemes where both displacements and thermal variables are solved either monolithically or in staggered fashion [42][43][44][45][46][47]. Traditionally, authors prefer the use of the temperature θ as the thermal unknown to be solved instead of the entropy density η (although the latter is equally plausible [42,48]). ...
... Insofar as the mixed-based system {p, F , H, J, E} (4) (or its alternative set {p, F , H, J, η}) is rather large, it will only be suitable to employ an explicit type of time integrator. In this work, an explicit one-step two-stage Total Variation Diminishing Runge-Kutta (TVD-RK) scheme is utilised [1,2,43,58,59,61,63,68,69]. This is described by the following time update equations from time step t n to t n+1 U a = U n a + ∆tU n a (U n a , t n ); U a = U a + ∆tU a (U a , t n+1 ); ...
... Additionally, initial conditions for the velocity, the triplet of deformation measures and temperature (see Figure 2 Figure 3 displays the spatial order of convergence of the numerical SPH approximation against the closed-form solution. For completeness, the spatial convergence of the numerical approximation using an in-house Upwind Vertex Centred Finite Volume Method (Upwind-VCFVM) [43] is also depicted. Both SPH and vertex centred implementations use the same Riemann based numerical dissipation. ...
Article
Full-text available
This paper presents a novel Smooth Particle Hydrodynamics computational framework for the simulation of large strain fast solid dynamics in thermo-elasticity. The formulation is based on the Total Lagrangian description of a system of first order conservation laws written in terms of the linear momentum, the triplet of deformation measures (also known as minors of the deformation gradient tensor) and the total energy of the system, extending thus the previous work carried out by some of the authors in the context of isothermal elasticity and elasto-plasticity. To ensure the stability (i.e. hyperbolicity) of the formulation from the continuum point of view, the internal energy density is expressed as a polyconvex combination of the triplet of deformation measures and the entropy density. Moreover, and to guarantee stability from the spatial discretisation point of view, consistently derived Riemann-based numerical dissipation is carefully introduced where local numerical entropy production is demonstrated via a novel technique in terms of the time rate of the so-called ballistic free energy of the system. For completeness, an alternative and equally competitive formulation (in the case of smooth solutions), expressed in terms of the entropy density, is also implemented and compared. A series of numerical examples is presented in order to assess the applicability and robustness of the proposed formulations, where the Smooth Particle Hydrodynamics scheme is benchmarked against an alternative in-house Finite Volume Vertex Centred implementation.
... The conservation of linear momentum per unit undeformed volume p = ρ 0 v (with ρ 0 the material density of the continuum) [29][30][31][78][79][80][81][82][83][84][85][86][87] is established for any arbitrary Lagrangian material volume 0 by ...
... where f 0 is a body force per unit of undeformed volume and t = P N is the traction vector associated with the material outward unit normal surface vector N with P being the first Piola-Kirchhoff stress tensor. The equivalent local equilibrium equation and the corresponding jump condition across a discontinuity [78,[82][83][84] can be written as ...
... Similar first-order conservation laws have been recently exploited in the field of solid mechanics[29][30][31][78][79][80][81][82][83][84][85][86][87]. ...
Article
Full-text available
This paper presents a new Smooth Particle Hydrodynamics computational framework for the solution of inviscid free surface flow problems. The formulation is based on the Total Lagrangian description of a system of first-order conservation laws written in terms of the linear momentum and the Jacobian of the deformation. One of the aims of this paper is to explore the use of Total Lagrangian description in the case of large deformations but without topological changes. In this case, the evaluation of spatial integrals is carried out with respect to the initial undeformed configuration, yielding an extremely efficient formulation where the need for continuous particle neighbouring search is completely circumvented. To guarantee stability from the SPH discretisation point of view, consistently derived Riemann-based numerical dissipation is suitably introduced where global numerical entropy production is demonstrated via a novel technique in terms of the time rate of the Hamiltonian of the system. Since the kernel derivatives presented in this work are fixed in the reference configuration, the non-physical clumping mechanism is completely removed. To fulfil conservation of the global angular momentum, a posteriori (least-squares) projection procedure is introduced. Finally, a wide spectrum of dedicated prototype problems is thoroughly examined. Through these tests, the SPH methodology overcomes by construction a number of persistent numerical drawbacks (e.g. hour-glassing, pressure instability, global conservation and/or completeness issues) commonly found in SPH literature, without resorting to the use of any ad-hoc user-defined artificial stabilisation parameters. Crucially, the overall SPH algorithm yields equal second order of convergence for both velocities and pressure.
... In recent years, some of the authors of this manuscript have pursued the same { p, F} system whilst exploiting well-established fluid inspired spatial discretisation techniques [3,[31][32][33][34]. In subsequent papers, the { p, F} system was then augmented by incorporating a new conservation law for the Jacobian of the deformation J [35] to effectively solve nearly/truly incompressible deformations. ...
... Consideration of thermal effects, especially in the context of large strain fast transient dynamics, is fundamental in order to obtain a realistic representation of stresses when a solid undergoes a complex and rapidly evolving deformation pattern. With focus on thermo-elasticity and thermo-inelasticity, numerous authors have worked over the years putting forward a variety of computational schemes where both displacements and thermal variables are solved either monolithically or in staggered fashion [33,[36][37][38][39][40]. Traditionally, authors prefer the use of the temperature θ as the thermal unknown to be solved, although alternative schemes in terms of the entropy η are also possible [36,41]. ...
... Comparing this with Eq. (40) at J = 1, the parameter Γ 0 can then be related to the linear thermal expansion as Γ 0 = 3α ∆t κ/c v . 10 For the limiting case of q = 0, the thermo-mechanical coupling term (42) becomes −η R (J ) = −c v Γ 0 ln J, which indeed is the expression used for perfect gases [33]. which gives a universally polyconvex strain energy function for Mie-Grüneisen model as ...
Article
In Parts I (Bonet et al., 2015) and II (Gil et al., 2016) of this series, a novel computational framework was presented for the numerical analysis of large strain fast solid dynamics in compressible and nearly/truly incompressible isothermal hyperelasticity. The methodology exploited the use of a system of first order Total Lagrangian conservation laws formulated in terms of the linear momentum and a triplet of deformation measures comprised of the deformation gradient tensor, its co-factor and its Jacobian. Moreover, the consideration of polyconvex constitutive laws was exploited in order to guarantee the hyperbolicity of the system and show the existence of a convex entropy function (sum of kinetic and strain energy per unit undeformed volume) necessary for symmetrisation. In this new paper, the framework is extended to the more general case of thermo-elasticity by incorporating the first law of thermodynamics as an additional conservation law, written in terms of either the entropy (suitable for smooth solutions) or the total energy density (suitable for discontinuous solutions) of the system. The paper is further enhanced with the following key novelties. First, sufficient conditions are put forward in terms of the internal energy density and the entropy measured at reference temperature in order to ensure ab-initio the polyconvexity of the internal energy density in terms of the extended set comprised of the triplet of deformation measures and the entropy. Second, the study of the eigenvalue structure of the system is performed as proof of hyperbolicity and with the purpose of obtaining correct time step bounds for explicit time integrators. Application to two well-established thermo-elastic models is presented: Mie–Grüneisen and modified entropic elasticity. Third, the use of polyconvex internal energy constitutive laws enables the definition of a generalised convex entropy function, namely the ballistic energy, and associated entropy fluxes, allowing the symmetrisation of the system of conservation laws in terms of entropy-conjugate fields. Fourth, and in line with the previous papers of the series, an explicit stabilised Petrov–Galerkin framework is presented for the numerical solution of the thermo-elastic system of conservation laws when considering the entropy as an unknown of the system. Finally, a series of numerical examples is presented in order to assess the applicability and robustness of the proposed formulation.
... In recent years, some of the authors of this manuscript have pursued the same {p, F } system whilst exploiting well-established fluid inspired spatial discretisation techniques [3,[31][32][33][34]. In subsequent papers, the {p, F } system was then augmented by incorporating a new conservation law for the Jacobian of the deformation J [35] to effectively solve nearly/truly incompressible deformations. ...
... Consideration of thermal effects, especially in the context of large strain fast transient dynamics, is fundamental in order to obtain a realistic representation of stresses when a solid undergoes a complex and rapidly evolving deformation pattern. With focus on thermo-elasticity and thermo-inelasticity, numerous authors have worked over the years putting forward a variety of computational schemes where both displacements and thermal variables are solved either monolithically or in staggered fashion [33,[36][37][38][39][40]. Traditionally, authors prefer the use of the temperature θ as the thermal unknown to be solved, although alternative schemes in terms of the entropy η are also possible [36,41]. ...
... which indeed is the expression used for perfect gases [33]. ...
... On the one hand, a new rst-order mixed form of the equations of nite strain solid dynamics is presented in [17,18,19,20]. In these works, the authors propose to use as primary variables the linear momentum p and the deformation gradient F. In order to eectively solve bending dominated scenarios in nearly incompressible cases they consider the introduction of the jacobian J as an extra unknown [21,22,23]. In more recent works [24,25,26], they insert the cofactor tensor of the deformation gradient H = cof F as an additional primary variable. ...
... It is interesting to observe that in the incompressible limit, when Poisson's ratio ν → 0.5 (for isotropic materials) then κ → ∞ and equation (22) reduces automatically to (23) ...
... For all the volumetric strain energy functions presented here, equation (23) imposes directly that J = 1 which is in fact the condition that a material must satisfy to be incompressible in nite strains. ...
Preprint
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In this work a new methodology for both the nearly and fully incompressible transient nite strain solid mechanics problem is presented. To this end, the momentum equation is complemented with a constitutive law for the pressure which emerges from the deviatoric/volumetric decomposition of the strain energy function for any hyperelas-tic material model. The incompressible limit is attained automatically depending on the material bulk modulus. The system is stabilized by means of the Variational Multiscale-Orthogonal Subgrid Scale method based on the decomposition of the unknowns into re-solvable and subgrid scales in order to prevent pressure uctuations. Several numerical examples are presented to assess the robustness and applicability of the proposed formulation .
... Recent developments in computational methods for fast solid dynamics [1,2,[7][8][9]18,19,21,[23][24][25] recommend the representation of motion and deformation of a given body via a system of first-order, mixed formulation conservation laws. The partial differential equations (PDEs) that form this system do not present the displacement as the main unknown to be evaluated, instead yielding a set of other relevant quantities (i. ...
... To date, the application of mixed formulation has been largely focused on nearly incompressible hyperelasticity [1,2,21,25], where many of the aforementioned shortcomings are encountered while using linear finite element methods. The work presented in this paper aims to investigate the feasibility of using the developed mixed formulation technique in the field of metal plasticity, by exploring some applications involving large material deformations. ...
... In (3.13), c p is the pressure wave speed, x min is the particle spacing, κ (2) and κ (4) are user-defined parameters, while the∇ 2 0 symbol represents a corrected Laplacian operator, applied to kernel W as detailed in [10]. ...
Article
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Due to its simplicity and robustness, smooth particle hydrodynamics (SPH) has been widely used in the modelling of solid and fluid mechanics problems. Through the years, various formulations and stabilisation techniques have been adopted to enhance it. Recently, the authors developed JST–SPH, a mixed formulation based on the SPH method. Originally devised for modelling (nearly) incompressible hyperelasticity, the JST–SPH formulation is mixed in the sense that linear momentum and a number of strain definitions, instead of the displacements, act as main unknowns of the problem. The resulting governing system of conservation laws conveniently enables the application of the Jameson–Schmidt–Turkel (JST) artificial dissipation term, commonly employed in computational fluid dynamics, to solid mechanics. Coupled with meshless SPH discretisation, this novel scheme eliminates the shortcomings encountered when implementing fast dynamics explicit codes using traditional mesh-based methods. This paper focuses on the applicability of the JST–SPH mixed formulation to the simulation of high-rate, large metal elastic–plastic deformations. Three applications—including the simulation of an industry-relevant metal forming process—are examined under different loading conditions, in order to demonstrate the reliability of the method. Results compare favourably with both data from the previous literature, and simulations performed with a commercial finite elements package. Most noticeably, these results demonstrate that the total Lagrangian framework of JST–SPH, fundamental to reduce the computational effort associated with the scheme, retains its accuracy in the presence of large distortions. Moreover, an algorithmic flow chart is included at the end of this document, to facilitate the computer implementation of the scheme.
... These polynomial equations are discontinuous across the element surface and as such a Riemann problem must be solved on the surface of the elements. The Riemann problem is a function of the jump in the shock impedance, velocity, and stress at the element surface and is identical to the one used in finite volume cell-centered hydrodynamic (CCH) methods [7][8][9][10]. The accuracy of a DG method is controlled by the order of the polynomial, where a linear polynomial P 1 produces up to second-order accuracy and a quadratic polynomial P 2 produces up to third-order accuracy. ...
... This prior work did not consider 2D higher-order elements or supporting hyperelastic-plastic models. Dynamic flows with hyperelastic-plastic models have been investigated using finite volume methods by Aguirre et al. [10]. The finite element (FE) method is widely used to solve the partial differential equations in complex physical systems with solid mechanics. ...
... A suite of 2D test problems are used to show the robustness and accuracy of the DG method with solid dynamics models as well as to show that the DG equation for evolving the deformation gradient behaves properly in multiple dimensions. The test suite used here follows previous works including those of Burton et al. [43], Aguirre et al. [10], Boscheri et al. [44] and Cheng et al. [45]. The first test problem is a bending elastic beam (or plate) using the finite strain hyperelastic model, which is a strong test of the stability and accuracy of the method because the problem should be able to run for long time periods without going unstable or dissipating the vibration behavior. ...
Article
We present a new multidimensional high-order Lagrangian discontinuous Galerkin (DG) hydrodynamic method that supports hypoelastic and hyperelastic strength models for simulating solid dynamics with higher-order elements. We also present new one-dimensional test problems that have an analytic solution corresponding to a hyperelastic–plastic wave. A modal DG approach is used to evolve fields relevant to conservation laws. These fields are approximated high-order Taylor series polynomials. The stress fields are represented using nodal quantities. The constitutive models used to calculate the deviatoric stress are either a hypoelastic–plastic, infinitesimal strain hyperelastic–plastic, or finite strain hyperelastic–plastic model. These constitutive models require new methods for calculating high-order polynomials for the velocity gradient and deformation gradient in an element. The plasticity associated with the strength model is determined using a radial return method with a J2 yield criterion and perfect plasticity. The temporal evolution of the governing equations is achieved with the total variation diminishing Runge–Kutta (TVD RK) time integration method. A diverse suite of 1D and 2D test problems are calculated. The new 1D piston test problems, which have analytic solutions for each elastic–plastic model, are presented and calculated to demonstrate the stability and formal accuracy of the various models with the new Lagrangian DG method. 2D test problems are calculated to demonstrate the stability and robustness of the new Lagrangian DG method on multidimensional problems with high-order elements, which have faces that can bend.
... In recent years, some of the authors of this manuscript have pursued the same {p, F } system whilst exploiting a wide range of spatial discretisation techniques. These include upwind cell centred FVM [43,44], Jameson-Schmidt-Turkel vertex centred FVM [45], upwind vertex centred FVM [46], two step Taylor-Galerkin FEM [47], stabilised Petrov-Galerkin FEM [48][49][50][51], Jameson-Schmidt-Turkel Smooth Particle Hydrodynamics (SPH) [52], Streamline Upwind Petrov-Galerkin SPH [53] and upwind SPH [54]. ...
... In subsequent papers, the {p, F } system was then augmented by incorporating a new conser-vation law for the Jacobian of the deformation J [46,49] to effectively solve nearly incompressible deformations. Moreover, the {p, F , J} formulation was also extended to account for truly incompressible materials utilising a tailor-made fractional step approach [49]. ...
... One contribution of the current paper is to enhance the robustness of the {p, F } vertex centred finite volume algorithm presented in [45,46] and to extend their applicability to nearly and truly incompressible scenarios. To achieve this, and following the work of [52,53,55,56], we incorporate another two additional geometric conservation laws, one for the co-factor of the deformation H (or area map) and the other for the Jacobian of the deformation J (or volume map). ...
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This paper presents an explicit vertex centred finite volume method for the solution of fast transient isothermal large strain solid dynamics via a system of first order hyperbolic conservation laws. Building upon previous work developed by the authors, in the context of alternative numerical discretisations, this paper explores the use of a series of enhancements (both from the formulation and numerical standpoints) in order to explore some limiting scenarios, such as the consideration of near and true incompressibility. Both Total and Updated Lagrangian formulations are presented and compared at the discrete level, where very small differences between both descriptions are observed due to the excellent discrete satisfaction of the involutions. In addition, a matrix-free tailor-made artificial compressibility algorithm is discussed and combined with an angular momentum projection algorithm. A wide spectrum of numerical examples is thoroughly examined. The scheme shows excellent (stable, consistent and accurate) behaviour, in comparison with other methodologies, in compressible, nearly incompressible and truly incompressible bending dominated scenarios, yielding equal second order of convergence for velocities, deviatoric and volumetric components of the stress.
... To address these shortcomings identified above, a novel mixed-based methodology tailor-made for emerging (industrial) solid mechanics problems has been recently proposed [1,2,3,7,8,9,10,11,12]. The mixedbased approach is written in the form of a system of first order hyperbolic conservation laws. ...
... Consider the three dimensional deformation of an elastic body moving from its initial configuration occupying a volume Ω 0 , of boundary ∂Ω 0 , to a current configuration at time t occupying a volume Ω, of boundary ∂Ω. The motion is defined through a deformation mapping x = φ(X, t) which satisfies the following set of mixed-based Total Lagrangian conservation laws [1,7,8,9,10,11,12] ...
... A standard benchmark problem of a twisting column is considered (see References [2,3,8,10,12] for details). The unit squared cross section column is twisted with a sinusoidal angular velocity field given by ω 0 = Ω[0, sin(πY /2H), 0] T rad/s, where Ω = 105 rad/s is the initial angular velocity and H = 6 m is the height of column. ...
Conference Paper
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An industry-driven computational framework for the numerical simulation of large strain explicit solid dynamics is presented. This work focuses on the spatial discretisation of a system of first order hyperbolic conservation laws using the cell centred Finite Volume Method [1, 2, 3]. The proposed methodology has been implemented as a parallelised explicit solid dynamics tool-kit within the CFD-based open-source platform OpenFOAM. Crucially, the proposed framework bridges the gap between Computational Fluid Dynamics and large strain solid dynamics. A wide spectrum of challenging numerical examples are examined in order to assess the robustness and parallel performance of the proposed solver.
... Specifically, a characteristic-based Riemann solver in conjunction with a linear reconstruction procedure is used, with the aim to guarantee both consistency and conservation of the overall algorithm. We show that the proposed SPH formulation is very similar in nature to that of the upwind vertex centred Finite Volume Method presented in [3]. In order to extend the application range towards the incompressibility limit, an artificial compressibility algorithm is also developed. ...
... The mixed-based set of equations is advanced in time by means of an explicit time integrator, where the time step is controlled through the Courant-Friedrichs-Lewy number [28] dependent on the volumetric wave speed c p . We also show that the new SPH algorithm is very similar to an upwind vertex centred FVM [3], where the latter requires the generation of a dual mesh using the medial dual approach [29]. In the near (or full) incompressibility limit, the wave speed c p can reach very high (even infinite in the degenerate case) values potentially leading to a very inefficient algorithm [26,30]. ...
... One of the key contributions of this paper is to show a relationship between the SPH particle approximation and the vertex centred Finite Volume Method (FVM) approximation [3], where the latter requires the definition of a dual mesh 3 which is constructed using the median dual approach [29]. This will enable a well-established upwinding finite volume spatial discretisation [49,50] commonly used in the CFD community to be adapted to a SPH mesh-free method. ...
Article
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In previous work (Lee et al., 2016, 2017), Lee et al. introduced a new Smooth Particle Hydrodynamics (SPH) computational framework for large strain explicit solid dynamics with special emphasis on the treatment of near incompressibility. A first order system of hyperbolic equations was presented expressed in terms of the linear momentum and the minors of the deformation, namely the deformation gradient, its co-factor and its Jacobian. Taking advantage of this representation, the suppression of numerical deficiencies (e.g. spurious pressure, long term instability and/or consistency issues) was addressed through well-established stabilisation procedures. In Reference Lee et al. (2016), the adaptation of the very efficient Jameson-Schmidt-Turkel algorithm was presented. Reference Lee et al. (2017) introduced an adapted variationally consistent Streamline Upwind Petrov Galerkin methodology. In this paper, we now introduce a third alternative stabilisation strategy, extremely competitive, and which does not require the selection of any user-defined artificial stabilisation parameter. Specifically, a characteristic-based Riemann solver in conjunction with a linear reconstruction procedure is used, with the aim to guarantee both consistency and conservation of the overall algorithm. We show that the proposed SPH formulation is very similar in nature to that of the upwind vertex centred Finite Volume Method presented in Aguirre et al. (2015). In order to extend the application range towards the incompressibility limit, an artificial compressibility algorithm is also developed. Finally, an extensive set of challenging numerical examples is analysed. The new SPH algorithm shows excellent behaviour in compressible, nearly incompressible and truly incompressible scenarios, yielding second order of convergence for velocities, deviatoric and volumetric components of the stress.
... Subsequently, the method has been extended to unstructured 3-D grids in a variety of forms, differing in terms of grid arrangements, discretisations, and primitive variables , for example, see Figure 8. Although cell-centred formulations are the most common form of Godunovtype method, vertex-centred [330,331] and, recently, face-centred approaches [346,347] have also been explored. A distinctive characteristic of Godunov-type methods is the adoption of fully explicit (a) 2-D unstructured cell-centred polygonal mesh from Kluth and Després [326] (b) 2-D/3-D unstructured vertex-centred polygonal/polyhedral mesh from Aguirre et al. [331] Figure 8. Forms of mesh employed by Kluth and Després [326] and Aguirre et al. [331] solution algorithms, where the time increment size is restricted by the standard Courant-Friedrichs-Lewy constraint [424]. ...
... Although cell-centred formulations are the most common form of Godunovtype method, vertex-centred [330,331] and, recently, face-centred approaches [346,347] have also been explored. A distinctive characteristic of Godunov-type methods is the adoption of fully explicit (a) 2-D unstructured cell-centred polygonal mesh from Kluth and Després [326] (b) 2-D/3-D unstructured vertex-centred polygonal/polyhedral mesh from Aguirre et al. [331] Figure 8. Forms of mesh employed by Kluth and Després [326] and Aguirre et al. [331] solution algorithms, where the time increment size is restricted by the standard Courant-Friedrichs-Lewy constraint [424]. Consequently, the approach is efficient for transient hyperbolic-type systems, and somewhat less so for elliptic ones, for example, quasi-static stress analysis. ...
... Although cell-centred formulations are the most common form of Godunovtype method, vertex-centred [330,331] and, recently, face-centred approaches [346,347] have also been explored. A distinctive characteristic of Godunov-type methods is the adoption of fully explicit (a) 2-D unstructured cell-centred polygonal mesh from Kluth and Després [326] (b) 2-D/3-D unstructured vertex-centred polygonal/polyhedral mesh from Aguirre et al. [331] Figure 8. Forms of mesh employed by Kluth and Després [326] and Aguirre et al. [331] solution algorithms, where the time increment size is restricted by the standard Courant-Friedrichs-Lewy constraint [424]. Consequently, the approach is efficient for transient hyperbolic-type systems, and somewhat less so for elliptic ones, for example, quasi-static stress analysis. ...
Preprint
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Since early publications in the late 1980s and early 1990s, the finite volume method has been shown suitable for solid mechanics analyses. At present, there are several flavours of the method, including `cell-centre', `staggered', `vertex-centred', `periodic heterogenous microstructural', `Godunov-type', `matrix-free', `meshless', as well as others. This article gives an overview, historical perspective, comparison and critical analysis of the different approaches, including their relative strengths, weaknesses, similarities and dissimilarities, where a close comparison with the de facto standard for computational solid mechanics, the finite element method, is given. The article finishes with a look towards future research directions and steps required for finite volume solid mechanics to achieve widespread acceptance.
... This approach has been studied using a wide variety of second order spatial discretisation techniques, well known in the CFD community. Specifically, two dimensional cell centred upwind FVM (2D upwind-CCFVM) [6], two-step Taylor-Galerkin FEM (2D TG-FEM) [14], vertex centred Jameson-Schmidt-Turkel FVM (3D JST-VCFVM) [11], stabilised Petrov-Galerkin FEM (3D PG-FEM) [16], vertex centred upwind FVM (3D upwind-VCFVM) [12], three dimensional cell centred upwind FVM (3D upwind-CCFVM) [8,9], Jameson-Schmidt-Turkel SPH (3D JST-SPH) [19] and Streamline Upwind Smooth Petrov-Galerkin SPH (3D SUPG-SPH) [20]. ...
... Subsequently, the two-field {p, F } formulation was then augmented by considering a new conservation law for the Jacobian J of the deformation to effectively solve nearly incompressible and truly incompressible materials [12,15]. Further enhancement of this {p, F , J} framework has recently been reported in [17,18], when considering compressible, nearly incompressible and truly incompressible materials governed by a polyconvex constitutive law, where the co-factor H of the deformation plays a dominant role, leading to a {p, F , H, J} system of conservation laws. ...
... In this study, consideration of irreversible processes is restricted to isothermal elasto-plastic materials typically used in metal forming applications [110]. In this particular work, thermal effects will be neglected (refer to [12] for the consideration of thermal effects). In order to model irrecoverable plastic behaviour, the standard rate-independent von Mises plasticity with isotropic hardening is used 10 . ...
Thesis
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In practical engineering applications involving extremely complex geometries, meshing typically constitutes a large portion of the overall design and analysis time. In the computational mechanics community, the ability to perform calculations on tetrahedral meshes has become increasingly important. For these reasons, automated tetrahedral mesh generation by means of Delaunay and advancing front techniques has recently received increasing attention in a number of applications, namely: crash simulations, cardiovascular modelling, blast and fracture modelling. Unfortunately, modern industry codes in solid mechanics (e.g. LS-DYNA, ANSYS AUTODYN, ABAQUS/Explicit, Altair Hypercrash) typically rely on the use of traditional displacement based Finite Element formulations which possess several distinct disadvantages, namely: (1) reduced order of convergence for strains and stresses in comparison to displacements; (2) high frequency noise in the vicinity of shocks; and (3) numerical instabilities associated with shear locking, volumetric locking and pressure checker-boarding. In order to address the above mentioned shortcomings, a new mixed-based set of equations for solid dynamics formulated in a system of first order hyperbolic conservation laws was introduced. The new set of conservation laws has a similar structure to that of the well known Euler equations in the context of Computational Fluid Dynamics (CFD). This enables us to borrow some of the available CFD technologies and to adapt the method in the context of solid dynamics. This thesis builds on the work carried out in Lee et. al. 2013 by further developing the upwind cell centred finite volume framework for the numerical analysis of large strain explicit solid dynamics and its tailor-made implementation within the open source code OpenFOAM, extensively used in industrial and academic environments. The object oriented nature of OpenFOAM implementation provides a very efficient platform for future development. In this computational framework, the primary unknown variables are linear momentum and deformation gradient tensor of the system. Moreover, the formulation is further extended for an additional set of geometric strain measures comprising of the co-factor of deformation gradient tensor and the Jacobian of deformation, in order to simulate polyconvex constitutive models ensuring material stability. The domain is spatially discretised using a standard Godunov-type cell centred framework where second order accuracy is achieved by employing a linear reconstruction procedure in conjunction with a slope limiter. This leads to discontinuities in variables at the cell interface which motivate the use of a Riemann solver by introducing an upwind bias into the evaluation of numerical contact fluxes. The acoustic Riemann solver presented is further developed by applying preconditioned dissipation to improve its performance in the near incompressibility regime and extending its range to contact applications. Moreover, two evolutionary frameworks are proposed in this study to satisfy the underlying involutions (or compatibility conditions) of the system. Additionally, the spatial discretisation is also represented through a node-based cell centred finite volume framework for comparison purposes. From a temporal discretisation point of view, a two stage Total Variation Diminishing Runge-Kutta time integrator is employed to ensure second order accuracy. Additionally, inclusion of a posteriori global angular momentum projection procedure enables preservation of angular momentum of the system. Finally, benchmark numerical examples are simulated to demonstrate mesh convergence, momentum preservation and the locking-free nature of the formulation. Moreover, the robustness and accuracy of the computational framework has been thoroughly examined through a series of challenging numerical examples involving contact scenarios and complex computational domains.
... Extension to consider path-dependent constitutive models will be explored in Section 4. Interestingly, through the imposition of suitable kinematic conditions [26], the ALE system (15) can degenerate into three alternative systems of first-order conservation equations. As shown in Table 1, these formulations include the well-established Total Lagrangian formulation [27,29,30,37,41,[48][49][50][51][52][53][54][55][56][57][58][59][60], the Eulerian formulation and the recently proposed Updated Reference Lagrangian formulation [32,37], which incorporates the concept of incremental kinematics. Thus, the ALE system (15) emerges as an elegant generalisation of various existing continuum conservation laws descriptions. ...
... The vertex centred finite volume spatial discretisation presented in this work requires the introduction of a median dual mesh [41,55,62,63] for defining control volumes (see Fig. 2). With this in mind and employing Gauss divergence theorem, expression (15) can be spatially discretised over a fixed referential control volume Ω to give ...
... Besides the SGH schemes and the CCH schemes, the vertex-centered hydrodynamics (VCH) has attracted more and more interests in recent years, see [35][36][37][38]. In this kind of schemes, the conserved variables (the mass, the momentum, and the total energy) are all stored at the nodes of mesh. ...
... Morgan et al. [36] presented a vertex-centered Arbitrary Lagrangian-Eulerian hydrodynamic approach for tetrahedral meshes, and this ALE version was derived from the Lagrangian one in [35]. Aguirre et al. [37] developed an upwind vertex-centered finite volume solver for Lagrangian solid dynamics. Liu et al. [38] developed a vertexcentered DG method in the direct ALE form for compressible single-material flow, and the vertex control volumes of this scheme are constructed with curved edges. ...
... Specifically, and by adopting referential configuration as an intermediate configuration during the deformation process, an extra conservation equation corresponding to the first law of thermodynamics (written in terms of the entropy density of the system) is solved in addition to the conservation of linear momentum and the three incremental geometric conservation laws (measured from referential domain to spatial domain). Interestingly, the methodology can indeed be degenerated into either a mixed-based set of Total [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] or Updated Lagrangian formulations [47] provided certain conditions are met. One key aspect that requires careful consideration is the overall stability of the algorithm. ...
... Notice that, if we update {F χ , H χ , J χ } continuously throughout the time integration process, a purely Updated Lagrangian first-order system [47] of conservation laws is retrieved. On the other hand, the Total Lagrangian formulation [32][33][34][35][36][37][38][39][40][41][42][43][44][45] is recovered if {F χ , H χ , J χ } are strongly enforced at the origin (that is, the reference configuration coincides with the material configuration). Detailed explanation of the transformations between the various formulations can be found in Reference [1]. ...
... These are known as weak solutions and satisfy certain jump conditions across the moving discontinuities, known as shocks (Eringen and Suhubi 1975;Gurtin et al. 2010;Bonet et al. 2021). Motivated by improving numerical simulations, significant research has been recently devoted to re-expressing the fundamental equations of solid dynamics as a set of conservation laws for the linear momentum and the deformation (Lee et al. 2013(Lee et al. , 2014Aguirre et al. 2014;Bonet et al. 2015;Aguirre et al. 2015;Haider et al. 2017). These conservation laws lead to a system of first order differential equations for the velocity field and the deformation together with associated jump conditions across moving shocks. ...
... This equation simply states that the rate of change of G is given by the gradient of the velocities. Equation (10) is the small strain counterpart of the evolution equation for the deformation gradient described in references Lee et al. 2013Lee et al. , 2014Aguirre et al. 2014;Bonet et al. 2015;Aguirre et al. 2015;Haider et al. 2017). ...
Article
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This paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory.
... Similar to cell-centred methods, both implicit and explicit solution algorithms have been developed, although explicit vertex-centred methods have seen less development. Within the category of explicit vertex-centred methods, there exists a variety of sub-classes, including: so-called dual-time-stepping explicit methods, where the solution is calculated in an explicit manner and a linear system is not directly formed [248,249,277,279,297]; 237] similar to those seen in cell-centred approaches, as noted in Sect. 2.1; and the so-called grid method [307][308][309][310][311][312]. ...
... The third-order diffusion term also serves a purpose towards choice of implicit components within the segregated solution algorithm: this is discussed further below. Alternative forms of diffusion/ smoothing terms have been also been proposed, for example, the fourth-order Jameson-Schmidt-Turkel [404] term employed in Godunov-type approaches [31,237], which takes the form of a Laplacian of a Laplacian: (14) The scalar coefficient K JST gives the correct dimension to the dissipation as well as controlling its magnitude. ...
Article
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Since early publications in the late 1980s and early 1990s, the finite volume method has been shown suitable for solid mechanics analyses. At present, there are several flavours of the method, which can be classified in a variety of ways, such as grid arrangement (cell-centred vs. staggered vs. vertex-centred), solution algorithm (implicit vs. explicit), and stabilisa- tion strategy (Rhie–Chow vs. Jameson–Schmidt–Turkel vs. Godunov upwinding). This article gives an overview, historical perspective, comparison and critical analysis of the different approaches where a close comparison with the de facto standard for computational solid mechanics, the finite element method, is given. The article finishes with a look towards future research directions and steps required for finite volume solid mechanics to achieve more widespread acceptance.
... The conservation of linear momentum per unit undeformed volume p = ρ 0 v (with ρ 0 the material density of the continuum) [29][30][31][78][79][80][81][82][83][84][85][86][87] is established for any arbitrary Lagrangian material volume Ω 0 by ...
... where f 0 is a body force per unit of undeformed volume and t = P N is the traction vector associated with the material outward unit normal surface vector N with P being the first Piola-Kirchhoff stress tensor. The equivalent local equilibrium equation and the corresponding jump condition across a discontinuity [78,[82][83][84] can be written as ...
... 2,14,15,17,19,33 Both, cell-centred and vertex-centred, FV techniques require a reconstruction of the gradient of the solution to ensure second-order convergence of the unknown and first-order convergence of the fluxes. 1,3,11,12 The accuracy of the scheme is therefore dependent on the accuracy of the reconstruction technique, which in turns depends on the quality of the mesh. In particular, FV methods are known to suffer an important loss of accuracy, and sometimes even a loss of second-order convergence, when unstructured meshes are used with highly stretched and/or deformed cells. ...
... The boundary conditions and source term are selected such that the analytical solution of the problem is known and given by The results of the analogous study in three dimensions are shown in Figure 3, demonstrating the optimal convergence of the error for both the solution and its gradient as well as the increased accuracy with respect to the first-order FCFV. In this case uniform tetrahedral meshes are used and the boundary conditions and source term are selected so that the analytical solution is known and given by u ex (x) = exp α sin(ax 1 + cx 2 + ex 3 ) + β cos(bx 1 ...
Article
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A second-order face-centred finite volume method (FCFV) is proposed. Contrary to the more popular cell-centred and vertex-centred finite volume (FV) techniques, the proposed method defines the solution on the faces of the mesh (edges in two dimensions). The method is based on a mixed formulation and therefore considers the solution and its gradient as independent unknowns. They are computed solving a cell-by-cell problem after the solution at the faces is determined. The proposed approach avoids the need of reconstructing the solution gradient, as required by cell-centred and vertex-centred FV methods. This strategy leads to a method that is insensitive to mesh distortion and stretching. The current method is second-order and requires the solution of a global system of equations of identical size and identical number of non-zero elements when compared to the recently proposed first-order FCFV. The formulation is presented for Poisson and Stokes problems. Numerical examples are used to illustrate the approximation properties of the method as well as to demonstrate its potential in three dimensional problems with complex geometries. The integration of a mesh adaptive procedure in the FCFV solution algorithm is also presented.
... On the one hand, a new first-order mixed form of the equations of finite strain solid dynamics is presented in [17,18,19,20]. In these works, the authors propose to use as primary variables the linear momentum p and the deformation gradient F. In order to effectively solve bending dominated scenarios in nearly incompressible cases they consider the introduction of the jacobian J as an extra unknown [21,22,23]. In more recent works [24,25,26], they insert the cofactor tensor of the deformation gradient H = cof F as an additional primary variable. ...
... For all the volumetric strain energy functions presented here, equation (23) imposes directly that J = 1 which is in fact the condition that a material must satisfy to be incompressible in finite strains. ...
Preprint
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In this work a new methodology for both the nearly and fully incompressible transient nite strain solid mechanics problem is presented. To this end, the momentum equation is complemented with a constitutive law for the pressure which emerges from the deviatoric/volumetric decomposition of the strain energy function for any hyperelas-tic material model. The incompressible limit is attained automatically depending on the material bulk modulus. The system is stabilized by means of the Variational Multiscale-Orthogonal Subgrid Scale method based on the decomposition of the unknowns into re-solvable and subgrid scales in order to prevent pressure uctuations. Several numerical examples are presented to assess the robustness and applicability of the proposed formulation.
... This paper will extend the new conservation law formulation of solid dynamics presented by the authors in References [1,2,3] to the field of thermo-elasticity and thermo-plasticity. In order to account for thermal effects, the total energy conservation law (also known as first law of thermodynamics) will be incorporated to the set of physical laws used to describe a deformable solid in [1], namely the conservation of linear momentum, deformation gradient (fibre map), cofactor of the deformation gradient (area map) and Jacobian of the deformation gradient (volume map). ...
... This paper will extend the new conservation law formulation of solid dynamics presented by the authors in References [1,2,3] to the field of thermo-elasticity and thermo-plasticity. In order to account for thermal effects, the total energy conservation law (also known as first law of thermodynamics) will be incorporated to the set of physical laws used to describe a deformable solid in [1], namely the conservation of linear momentum, deformation gradient (fibre map), cofactor of the deformation gradient (area map) and Jacobian of the deformation gradient (volume map). ...
Conference Paper
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This paper will extend the new conservation law formulation of solid dynamics presented by the authors in References [1, 2, 3] to the field of thermo-elasticity and thermo-plasticity. In order to account for thermal effects, the total energy conservation law (also known as first law of thermodynamics) will be incorporated to the set of physical laws used to describe a deformable solid in [1], namely the conservation of linear momentum, deformation gradient (fibre map), co-factor of the deformation gradient (area map) and Jacobian of the deformation gradient (volume map). From the spatial discretisation viewpoint, a vertex centred finite volume method will be utilised. Naturally, discontinuity of the conservation variables across (dual) control volume interfaces leads to a Riemann problem, whose approximate solution will be derived by means of the (nonlinear) Riemann solver. Taking advantage of the conservation formulation for solids, well-established shock capturing algorithm will also be used to greatly enhance the performance of the algorithm in the vicinity of severe thermal shocks. The paper will provide examples based on Mie-Gruneisen equation of state and entropic elasticity. A number of benchmark tests will be provided in order to demonstrate the robustness and applicability of the proposed methodology. In this case, both stresses and temperature converge at the same rate as velocities and displacements. This is in clear contrast to the classical displacement based formulations where derived variables (e.g. stresses and strains) converge at one order below the rate of displacements. For completeness, comparisons with other in-house finite volume [3] or particle based methodologies [2] will be provided.
... The unknowns are the displacements, the stresses and the strains, although in some cases it is also convenient to introduce other intermediate variables, or parts of the stresses or of the strains, mainly their volumetric and deviatoric components; in particular, this is useful in the case of incompressible materials. [5][6][7][8] The geometric and the constitutive equations can be used to write Cauchy's equation in terms of the displacements only. 9 This is the so-called irreducible form of the problem, perhaps the most widely used. ...
Article
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This paper presents mixed finite element formulations to approximate the hyperelasticity problem using as unknowns the displacements and either stresses or pressure or both. These mixed formulations require either finite element spaces for the unknowns that satisfy the proper inf‐sup conditions to guarantee stability or to employ stabilized finite element formulations that provide freedom for the choice of the interpolating spaces. The latter approach is followed in this work, using the Variational Multiscale concept to derive these formulations. Regarding the tackling of the geometry, we consider both infinitesimal and finite strain problems, considering for the latter both an updated Lagrangian and a total Lagrangian description of the governing equations. The combination of the different geometrical descriptions and the mixed formulations employed provides a good number of alternatives that are all reviewed in this paper.
... Both material particle motion (79) and velocity (80) are carefully designed to ensure movement is constrained tangentially along the material boundary to prevent volume change. ...
Article
This paper introduces a novel Smooth Particle Hydrodynamics (SPH) computational framework that incorporates an Arbitrary Lagrangian Eulerian (ALE) formalism, expressed through a system of first-order conservation laws. In addition to the standard material and spatial configurations, an additional (fixed) referential configuration is introduced. The ALE conservative framework is established based on the fundamental conservation principles, including mass, linear momentum and the first law of thermodynamics represented through entropy density. A key contribution of this work lies in the evaluation of the physical deformation gradient tensor, which measures deformation from material to spatial configuration through a multiplicative decomposition into two auxiliary deformation gradient tensors. Both of the deformation tensors are obtained via additional first-order conservation equations. Interestingly, the new ALE conservative formulation will be shown to degenerate into alternative mixed systems of conservation laws for solid dynamics: particle-shifting, velocity-shifting and Eulerian formulations. The framework also considers path-and/or strain rate-dependent constitutive models, such as isothermal plasticity and thermo-visco-plasticity, by integrating evolution equations for internal state variables. Another contribution of this paper is the evaluation of ALE motion (known as smoothing procedure) by solving a conservation-type momentum equation. This procedure is indeed useful for maintaining a regular particle distribution and enhancing solution accuracy in regions characterised by large plastic flows. The hyperbolicity of the underlying system is ensured and accurate wave speed bounds in the context of ALE description are presented, crucial for ensuring the stability of explicit time integrators. For spatial discretisation, a Godunov-type SPH method is employed and adapted. To guarantee stability from the semi-discretisation standpoint, a carefully designed numerical stabilisation is introduced. The Lyapunov stability analysis is carried out by assessing the time rate of the Ballistic energy of the system, aiming to ensure non-negative entropy production. In order to ensure the global conservation of angular momentum, we employ a three-stage Runge-Kutta time integrator together with a discrete angular momentum projection algorithm. Finally, a range of three dimensional benchmark problems are examined to illustrate the robustness and applicability of the framework. The developed ALE SPH scheme outperforms the Total Lagrangian SPH framework, particularly excelling in capturing plasticity regimes with optimal computational efficiency.
... The vertex centred finite volume spatial discretisation presented in this work requires the introduction of a median dual mesh 41,51,64,65 for the definition of control volumes (see Figure 2). With this in mind, and making use of Gauss divergence theorem, expression (38) can now be spatially discretised over a fixed referential control volume Ω a , to give ...
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The paper introduces a computational framework using a novel Arbitrary Lagrangian Eulerian (ALE) formalism in the form of a system of first‐order conservation laws. In addition to the usual material and spatial configurations, an additional referential (intrinsic) configuration is introduced in order to disassociate material particles from mesh positions. Using isothermal hyperelasticity as a starting point, mass, linear momentum and total energy conservation equations are written and solved with respect to the reference configuration. In addition, with the purpose of guaranteeing equal order of convergence of strains/stresses and velocities/displacements, the computation of the standard deformation gradient tensor (measured from material to spatial configuration) is obtained via its multiplicative decomposition into two auxiliary deformation gradient tensors, both computed via additional first‐order conservation laws. Crucially, the new ALE conservative formulation will be shown to degenerate elegantly into alternative mixed systems of conservation laws such as Total Lagrangian, Eulerian and Updated Reference Lagrangian. Hyperbolicity of the system of conservation laws will be shown and the accurate wave speed bounds will be presented, the latter critical to ensure stability of explicit time integrators. For spatial discretisation, a vertex‐based Finite Volume method is employed and suitably adapted. To guarantee stability from both the continuum and the semi‐discretisation standpoints, an appropriate numerical interface flux (by means of the Rankine–Hugoniot jump conditions) is carefully designed and presented. Stability is demonstrated via the use of the time variation of the Hamiltonian of the system, seeking to ensure the positive production of numerical entropy. A range of three dimensional benchmark problems will be presented in order to demonstrate the robustness and reliability of the framework. Examples will be restricted to the case of isothermal reversible elasticity to demonstrate the potential of the new formulation.
... The problem to be solved consists essentially of the equation for the conservation of linear momentum (Cauchy's equation), the geometric equation relating strains and displacements and the constitutive equation relating stresses and strains. The unknowns are the displacements, the stresses and the strains, although in some cases it is also convenient to introduce other intermediate variables, or parts of the stresses or of the strains, mainly their volumetric and deviatoric components; in particular, this is useful in the case of incompressible materials [5,6,7,8]. ...
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This paper presents mixed finite element formulations to approximate the hyperelasticity problem using as unknowns the displacements and either stresses or pressure or both. These mixed formulations require either finite element spaces for the unknowns that satisfy the proper inf-sup conditions to guarantee stability or to employ stabilized finite element formulations that provide freedom for the choice of the interpolating spaces. The latter approach is followed in this work, using the Variational Multiscale concept to derive these formulations. Regarding the tackling of the geometry, we consider both infinitesimal and finite strain problems, considering for the latter both an updated Lagrangian and a total Lagrangian description of the governing equations. The combination of the different geometrical descriptions and the mixed formulations employed provides a good number of alternatives that are all reviewed in this paper.
... In [47,71,72] stabilizing mechanisms were introduced through a combination of SUPG stabilization [49-51, 55-60, 99, 100] and penalties on the difference between the linear and constant approximation of the deformation gradient (in analogy to Puso and Solberg [90]). Alternatively, Aguirre et al. [3,4] pursued stability by means of approximate Riemann solvers initially developed for nodal finite volumes. A recent work of Cardiff et al. [15] also explores these general ideas. ...
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We propose a stabilized linear tetrahedral finite element method for static, finite elasticity problems involving com-pressible and nearly incompressible materials. Our approach relies on a mixed formulation, in which the nodal displacement unknown filed is complemented by a nodal Jacobian determinant unknown field. This approach is simple to implement in practical applications (e.g., in commercial software), since it only requires information already available when computing the Newton-Raphson tangent matrix associated with irreducible (i.e., displacement-based) finite element formulations. By nature, the proposed method is easily extensible to nonlinear models involving visco-plastic flow. An extensive suite of numerical tests in two and three dimensions is presented, to demonstrate the performance of the method.
... Specifically, and by adopting referential configuration as an intermediate configuration during the deformation process, an extra conservation equation corresponding to the first law of thermodynamics (written in terms of the entropy density of the system) is solved in addition to the conservation of linear momentum and the three incremental geometric conservation laws (measured from referential domain to spatial domain). Interestingly, the methodology can indeed be degenerated into either a mixed-based set of Total [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] or Updated Lagrangian formulations [47] provided certain conditions are met. One key aspect that requires careful consideration is the overall stability of the algorithm. ...
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This paper introduces a novel upwind Updated Reference Lagrangian Smoothed Particle Hydrodynamics (SPH) algorithm for the numerical simulation of large strain thermo-elasticity and thermo-visco-plasticity. The deformation process is described via a system of first-order hyperbolic conservation laws expressed in referential description, chosen to be an intermediate configuration of the deformation. The linear momentum, the three incremental geometric strains measures (between referential and spatial domains), and the entropy density of the system are treated as conservation variables of this mixed coupled approach, thus extending the previous work of the authors in the context of isothermal elasticity and elasto-plasticity. To guarantee stability from the SPH discretisation standpoint, appropriate entropy-stable upwinding stabilisation is suitably designed and presented. This is demonstrated via the use of the Ballistic free energy of the coupled system (also known as Lyapunov function), to ensure the satisfaction of numerical entropy production. An extensive set of numerical examples is examined in order to assess the applicability and performance of the algorithm. It is shown that the overall algorithm eliminates the appearance of spurious modes (such as hour-glassing and non-physical pressure fluctuations) in the solution, typical limitations observed in the classical Updated Lagrangian SPH framework.
... The problem is stabilized with the VMS framework. A family of first-order form of the equations is presented in [27,28,29,30,31,32] where the authors propose to use as primary variables the linear momentum p, the deformation gradient F, the cofactor tensor of the deformation gradient H and the jacobian J; the objective for this choice of variables is to ease dealing with some complex constitutive laws, and in particular with polyconvex hyperelastic potentials. In [33] the incompressibility of the material is treated with the displacement/pressure pair in an updated Lagrangian formulation framework. ...
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In this work a new methodology for finite strain solid dynamics problems for stress accurate analysis including the incompressible limit is presented. In previous works, the authors have presented the stabilized mixed displacement/pressure formulation to deal with the incompressibility constraint in finite strain solid dynamics. To this end, the momentum equation is complemented with a constitutive law for the pressure which emerges from the deviatoric/volumetric decomposition of the strain energy function for any hyperelastic material model. The incompressible limit is attained automatically depending on the material bulk modulus. This work exploits the concept of mixed methods to formulate stable displacement/pressure/deviatoric stress finite elements. The final goal is to design a finite element technology able to tackle simultaneously problems which may involve incompressible behavior together with a high degree of accuracy of the stress field. The Variational Multi-Scale stabilization technique and, in particular, the Orthogonal Subgrid Scale method allows the use of equal-order interpolations. These stabilization procedures lead to discrete problems which are fully stable, free of volumetric locking, stress oscillations and pressure fluctuations. Numerical benchmarks show that the results obtained compare very favorably with those obtained with the corresponding stabilized mixed displacement/pressure formulation.
... The proposed variational approach is illustrated with thermo-hyperelastic-viscoplastic solid media, especially with the Johnson-Cook viscoplastic flow rule [39]. The main advantage of this new variational formulation is that it is naturally compatible with any numerical scheme dedicated to the approximation of the solution of hyperbolic systems, like finite volumes [25,48,1], particle methods [29], Discontinuous Galerkin approaches [20,16], or finite element method [12], written in conservation form. Here, the second order accurate flux difference splitting finite volume method [46,36] is used, which permits to solve the set of conservation laws on examples involving either small or large strains. ...
Article
In this paper, a variational framework is proposed for the constitutive update of thermomechanical constitutive models in the special case where their input results from quantities directly updated by hyperbolic conservation laws. Both a continuum and a consistent first order accurate discrete settings are derived. The originality of this work lies in that the constitutive update is driven by the rates of some strain measure and the internal energy density in the continuum setting, leading to a rate-type description of the local constitutive problem, and by the updated values at some discrete time of these strain measure and internal energy density in the discrete setting. These quantities are updated by the solution of a system of discrete conservation laws including the first principle of thermodynamics, ensuring that the right shock speeds will be computed. This point is of crucial importance when simulating impact on structures for instance. The proposed variational approach is illustrated for thermo-hyperelastic-viscoplastic solid media, especially using the parameterization of the flow rule direction based on pseudo-stresses proposed by Mosler & co-workers. The proposed discrete variational solver is then coupled with the second order accurate flux difference splitting finite volume method, which permits to solve the set of conservation laws. Comparisons are performed on a set of test cases with numerical solutions obtained with finite elements coupled to an explicit time-stepping and to a temperature-driven variational constitutive update. They allow to show the good behavior of the proposed approach.
... As the system of conservation laws presented above has more equations than needed, suitable compatibility relationships (also known as involutions [47,48,56,57]) are necessary, namely ...
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This paper presents a new Updated Reference Lagrangian Smooth Particle Hydrodynamics (SPH) algorithm for the analysis of large deformation isothermal elasticity and elasto-plasticity. Taking as point of departure a Total Lagrangian setting and considering as referential configuration an intermediate configuration of the deformation process, the equation of conservation of linear momentum and three geometric conservation laws (for the deformation gradient, its cofactor and its determinant) are rewritten leading to a very generic system of first order conservation laws. The key feature of the formulation is a suitable multiplicative decomposition of the conservation variables, leading to a very simple final set of equations with striking similarities to the conventional Total Lagrangian system albeit rewritten in terms of alternative Referential Updated conservation variables which are evolved in time. Taking advantage of this new Updated Reference Lagrangian formalism, a second order entropy-stable SPH upwiding stabilisation method will be introduced. With respect to previous publications by the group, a new three-stage Runge-Kutta time integration method is implemented in order to increase the CFL stability restriction. Finally, and to demonstrate the robustness and applicability of the methodology, a wide spectrum of challenging problems will be presented and compared, including some benchmark three-dimensional large deformation elasto-plasticity problems. To facilitate its ease of implementation, the paper explores the use of a series of novel expressions for the evaluation of kernels and the gradients of kernels to provide the SPH user the freedom to choose amongst various options, all leading to equally convincingly robust results.
... The results obtained are second order accurate in stress. Moreover, in the incompressible limit, and bending dominated problems, an additional variable was introduced, which is the Jacobian determinant of the deformation gradient J [26] [27]. In recent works [28][29] [30], a nodal co-factor tensor H = cof : F is added. ...
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This paper proposes a novel way to solve transient linear, and non-linear solid dynamics for compressible, nearly incompressible, and incompressible material in the updated Lagrangian framework for tetrahedral unstructured finite elements. It consists of a mixed formulation in both displacement and pressure, where the momentum equation of the continuum is complemented with a pressure equation that handles incompressibility inherently. It is obtained through the deviatoric and volumetric split of the stress, that enables us to solve the problem in the incompressible limit. A linearization of the deviatoric part of the stress is implemented as well. The Variational Multi-Scale method (VMS) is developed based on the orthogonal decomposition of the variables, which damps out spurious pressure fields for piece wise linear tetrahedral elements. Various numerical examples are presented to assess the robustness, accuracy and capabilities of our scheme in bending dominated problems, and for complex geometries.
... The results obtained are second order accurate in stress. Moreover, in the incompressible limit, and bending dominated problems, an additional variable was introduced, which is the Jacobian determinant of the deformation gradient J [26] [27]. In recent works [28][29] [30], a nodal co-factor tensor H = cof : F is added. ...
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This paper proposes a novel way to solve transient linear, and non-linear solid dynamics for compressible, nearly incompressible, and incompressible material in the updated Lagrangian framework for tetrahedral unstructured finite elements. It consists of a mixed formulation in both displacement and pressure, where the momentum equation of the continuum is complemented with a pressure equation that handles incompresibility inherently. It is obtained through the deviatoric and volumetric split of the stress, that enables us to solve the problem in the incompressible limit. The Varitaional Multi-Scale method (VMS) is developed based on the orthogonal decomposition of the variables, which damps out spurious pressure fields for piece wise linear tetrahedral elements. Various numerical examples are presented to assess the robustness, accuracy and capabilities of our scheme in bending dominated problems, and for complex geometries.
... On the one hand, a new first-order mixed form of the equations of finite strain solid dynamics is presented in [17][18][19][20]. In these works, the authors propose to use as primary variables the linear momentum p and the deformation gradient F. In order to effectively solve bending dominated scenarios in nearly incompressible cases they consider the introduction of the jacobian J as an extra unknown [21][22][23]. In more recent works [24][25][26], they insert the cofactor tensor of the deformation gradient H = cof F as an additional primary variable. ...
Article
In this work a new methodology for both the nearly and fully incompressible transient finite strain solid mechanics problem is presented. To this end, the momentum equation is complemented with a constitutive law for the pressure which emerges from the deviatoric/volumetric decomposition of the strain energy function for any hyperelastic material model. The incompressible limit is attained automatically depending on the material bulk modulus. The system is stabilized by means of the Variational Multiscale-Orthogonal Subgrid Scale method based on the decomposition of the unknowns into resolvable and subgrid scales in order to prevent pressure fluctuations. Several numerical examples are presented to assess the robustness and applicability of the proposed formulation.
... 2,14,15,17,19,33 Both, cell-centred and vertex-centred, FV techniques require a reconstruction of the gradient of the solution to ensure second-order convergence of the unknown and first-order convergence of the fluxes. 1,3,11,12 The accuracy of the scheme is therefore dependent on the accuracy of the reconstruction technique, which in turns depends on the quality of the mesh. In particular, FV methods are known to suffer an important loss of accuracy, and sometimes even a loss of second-order convergence, when unstructured meshes are used or highly stretched and deformed cells are present in the computational mesh. ...
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A second-order face-centred finite volume method (FCFV) is proposed. Contrary to the more popular cell-centred and vertex-centred finite volume (FV) techniques, the proposed method defines the solution on the faces of the mesh (edges in two dimensions). The method is based on a mixed formulation and therefore considers the solution and its gradient as independent unknowns. They are computed solving an element-by-element problem after the solution at the faces is determined. The proposed approach avoids the need of reconstructing the solution gradient, as required by cell-centred and vertex-centred FV methods. This strategy leads to a method that is insensitive to mesh distortion and stretching. The current method is second-order and requires the solution of a global system of equations of identical size and identical number of non-zero elements when compared to the recently proposed first-order FCFV. The formulation is presented for Poisson and Stokes problems. Numerical examples are used to illustrate the approximation properties of the method as well as to demonstrate its potential in three dimensional problems with complex geometries. The integration of a mesh adaptive procedure in the FCFV solution algorithm is also presented.
... Although finite volume technique is rather a new arrival as a numerical method for the solution of solid mechanics problems compared to the finite element method, however, it has shown promising capabilities in studies so far. Elastic analysis of three-dimensional solids (Bailey and Cross, 1995), stress analysis of elasto-plastic solids (Demirdzic and Martinovic, 1993), FV methods for different solid mechanics problems (Cardiff et al., 2017;Cardiff et al., 2016;Aguirre et al., 2015;Tang et al., 2015;Nordbotten, 2014;Trangenstein, 1991;Demirdzˇic ́ et al., 1988), FV-based flow formulations to simulate plastic deformation processes (Basic ́ et al., 2005;Trangenstein, 1991), bending analysis of elastic plates (Wheel, 1997;Fallah, 2004;Fallah, 2006), finite volume analysis of dynamic fracture problems (Ivankovic et al., 1994;Stylianou and Ivankovic, 2002) and studying periodic and functionally graded media (Cavalcante et al., 2011; are some of the works which have highlighted notable capabilities of the finite volume method. This paper extends the present author's work on the application of finite volume method for the elasto-plastic analysis of Mindlin plates in which the layered approach was adopted (Fallah and Parayandeh, 2014). ...
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This paper extends the previous work of authors and presents a non-layered Finite Volume formulation for the elasto-plastic analysis of Mindlin-Reissner plates. The incremental algorithm of the elasto-plastic solution procedure is shown in detail. The performance of the formulation is examined by analyzing of plates with different boundary conditions and loading types. The results are illustrated and compared with the predictions of the layered approach. These several comparisons reveal that the non-layered Finite Volume approach can present accurate results with low CPU time usage despite its simplicity of the solution procedure.
... The FV method applied to CSM comes in many forms, where the presented approach stems from the seminal work of Demirdžić et al. [6] and co-workers in the late 1980s. A large number of alternative forms are found in literature, including fully explicit approaches, vertex-centred methods and so-called parametric formulations [66,[70][71][72][73][74][104][105][106]. ...
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Over the past 30 years, the cell-centred finite volume method has developed to become a viable alternative to the finite element method in the field of computational solid mechanics. The current article presents an open-source toolbox for solid mechanics and fluid-solid interaction simulations based on the finite volume library OpenFOAM. The object-oriented toolbox design is outlined, where emphasis has been given to code use, comprehension, maintenance and extension. The toolbox capabilities are demonstrated on a number of representative test problems, where comparisons are given with finite element solutions.
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In this work a new methodology for finite strain solid dynamics problems for stress accurate analysis including the incompressible limit is presented. In previous works, the authors have presented the stabilized mixed displacement/pressure formulation to deal with the incompressibility constraint in finite strain solid dynamics. To this end, the momentum equation is complemented with a constitutive law for the pressure which emerges from the deviatoric/volumetric decomposition of the strain energy function for any hyperelastic material model. The incompressible limit is attained automatically depending on the material bulk modulus. This work exploits the concept of mixed methods to formulate stable displacement/pressure/deviatoric stress finite elements. The final goal is to design a finite element technology able to tackle simultaneously problems which may involve incompressible behavior together with a high degree of accuracy of the stress field. The Variational Multi‐Scale stabilization technique and, in particular, the Orthogonal Subgrid Scale method allows the use of equal‐order interpolations. These stabilization procedures lead to discrete problems which are fully stable, free of volumetric locking, stress oscillations and pressure fluctuations. Numerical benchmarks show that the results obtained compare very favorably with those obtained with the corresponding stabilized mixed displacement/pressure formulation.
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The need to simulate flexible, relatively thin structure is of growing interest with applications ranging from thin cylindrical sensors to membrane-like structures. These structures usually interact with their surroundings to accumulate data, or for a specific purpose. The inevitable interaction between the surrounding fluid and the solid is solved using a novel Fluid-Structure Interaction (FSI) coupling scheme. This thesis proposes a novel way to model the interaction between the fluid and solid. It consists of a hybrid method that combines both the traditional monolithic and partitioned approaches for Fluid-Structure Interaction (FSI). The solid mesh is immersed in a fluid-solid mesh at each time step, whilst having its own independent Lagrangian hyperelastic solver. The hyperelastic solver consists of a mixed formulation in both displacement and pressure, where the momentum equation of the continuum is complemented with a pressure equation that handles incompressibility inherently. It is obtained through the deviatoric and volumetric split of the stress that enables us to solve the problem in the incompressible limit. A linearization of the deviatoric part of the stress is implemented as well. The Eulerian mesh contains both the fluid and solid, and accommodates additional physical phenomena. Anisotropic mesh adaptation and the Level-Set methods are used for the interface coupling between the solid and fluid to better capture the interaction between them. All of the above components form the Adaptive Immersed Mesh Method (AIMM). The Variational Multi-Scale (VMS) method is used for both solvers to damp out any spurious oscillations that may arise for piece wise linear tetrahedral elements. The framework is constructed in 3D with parallel computing in mind. Extensive 2D and 3D test cases are presented that validate the hyperelastic Lagrangian solver, and the FSI AIMM framework. An application of the industrial partners was lastly tackled.
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This article introduces a new vertex-centered quasi-Lagrangian discontinuous Galerkin method for two-dimensional compressible flows that is third-order accurate both in space and time. The computational domain is divided into structured quadrilateral cells with straight edges. Nodal control volumes with curved edges are constructed surrounding the grid vertices. The Euler equations in arbitrary Lagrangian-Eulerian (ALE) form are discretized on these nodal control volumes using a discontinuous Galerkin method. The time marching is implemented by the third-order strong-stability-preserving Runge-Kutta method. A compact HWENO reconstruction algorithm is used as limiter to eliminate spurious oscillations near discontinuities. The polynomial expression of fluid velocity defined in a nodal control volume is also obtained from the reconstruction procedure, and is used to calculate the moving velocity of corresponding grid vertex. In this way, the grid vertices are moved in a rigorously Lagrangian manner, although there are still mass fluxes between neighbouring nodal control volumes. Therefore the scheme is called as a quasi-Lagrangian one. The scheme is conservative for mass, momentum and total energy. Some numerical tests are carried out to demonstrate the accuracy and robustness of the scheme. It has a favorable qualitative behavior for discontinuous problems and optimal convergence rates for smooth problems.
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The extension of the 3D cell-centered Finite Volume EUCCLHYD scheme to the hyperelasticity system is proposed here. This study is based on the left Cauchy-Green tensor B which enables to work in a fully updated Lagrangian formalism. The second order extension of this scheme is proposed using a MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws) procedure combined with a GRP (Generalized Riemann Problem) approach. In particular, the limitation of the tensor fields is done in a component-wise manner. Moreover, the complete GRP procedure is proposed in the case of Neo-Hookean compressible solids. The scheme is validated on several test cases introducing small as well as large deformations. In particular, good results are found on the non trivial problems of oscillating and twisting beams.
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Two formulations of the Lax-Wendroff scheme are proposed in this paper for its extension to elastic-plastic solids. The unknown vector consists of the strain or stress components in addition to the velocity ones according to the chosen formulation. The Lax-Wendroff scheme is here implemented in its Richtmyer two-step version so that the projection of the elastic trial stresses onto the yield locus is performed twice per time step. It is shown that it allows to reduce the number of cells required for the approximation of discontinuous plastic waves with respect to a classical one-step strategy consisting in an a posteriori projection of the elastic trial stresses onto the yield locus. Elastic-plastic associated and non-associated constitutive models under small strains are considered in examples. In particular, dynamic ratchetting is simulated with the Armstrong-Frederick nonlinear kinematic hardening. Comparisons with respect to finite element numerical solutions show good agreements.
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This paper introduces a new computational framework for the analysis of large strain fast solid dynamics. The paper builds upon previous work published by the authors (Gil et al., 2014) [1], where a first order system of hyperbolic equations is introduced for the simulation of isothermal elastic materials in terms of the linear momentum, the deformation gradient and its Jacobian as unknown variables. In this work, the formulation is further enhanced with four key novelties. First, the use of a new geometric conservation law for the co-factor of the deformation leads to an enhanced mixed formulation, advantageous in those scenarios where the co-factor plays a dominant role. Second, the use of polyconvex strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes for solid dynamics problems. Moreover, the introduction of suitable conjugate entropy variables enables the derivation of a symmetric system of hyperbolic equations, dual of that expressed in terms of conservation variables. Third, the new use of a tensor cross product (de Boer, 1982) greatly facilitates the algebraic manipulations of expressions involving the co-factor of the deformation. Fourth, the development of a stabilised Petrov–Galerkin framework is presented for both systems of hyperbolic equations, that is, when expressed in terms of either conservation or entropy variables. As an example, a polyconvex Mooney-Rivlin material is used and, for completeness, the eigen-structure of the resulting system of equations is studied to guarantee the existence of real wave speeds. Finally, a series of numerical examples is presented in order to assess the robustness and accuracy of the new mixed methodology, benchmarking it against an ample spectrum of alternative numerical strategies, including implicit multi-field Fraeijs de Veubeke-Hu-Washizu variational type approaches and explicit cell and vertex centred Finite Volume schemes.
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We consider the numerical simulation of inviscid reactive flows with application to high density explosive detonation. The numerical model is based on the Euler equations and the Mie-Grüneisen equation of state extended to treat chemical energy release and expanded states. The equations are computed with a Roe-Glaister solver on a Cartesian mesh. We present results for two substances, a binder and an explosive. Our solution method is verified against the exact solution of the shock tube problem for solid materials. We show under what conditions a "physical" expansion shock can appear in this example. We then address the problem of modeling expanded states, and show results for a two-dimensional shock distraction around a sharp corner. In the last part of the paper, we introduce a detonation model that extends the Mie-Grüneisen equation of state to enable high explosive simulations without the complexity of mixture equations of state. We conclude with two examples of corner-turning computations carried out with a pressure-dependent reaction rate law.
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SUMMARYA finite volume cell-centered Lagrangian hydrodynamics approach, formulated in Cartesian frame, is presented for solving elasto-plastic response of solids in general unstructured grids. Because solid materials can sustain significant shear deformation, evolution equations for stress and strain fields are solved in addition to mass, momentum, and energy conservation laws. The total stress is split into deviatoric shear stress and dilatational components. The dilatational response of the material is modeled using the Mie-Grüneisen equation of state. A predicted trial elastic deviatoric stress state is evolved assuming a pure elastic deformation in accordance with the hypo-elastic stress-strain relation. The evolution equations are advanced in time by constructing vertex velocity and corner traction force vectors using multi-dimensional Riemann solutions erected at mesh vertices. Conservation of momentum and total energy along with the increase in entropy principle are invoked for computing these quantities at the vertices. Final state of deviatoric stress is effected via radial return algorithm based on the J-2 von Mises yield condition. The scheme presented in this work is second-order accurate both in space and time. The suitability of the scheme is evinced by solving one- and two-dimensional benchmark problems both in structured grids and in unstructured grids with polygonal cells. Copyright © 2013 John Wiley & Sons, Ltd.
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Purpose – The purpose of this paper is to present a new stabilised low-order finite element methodology for large strain fast dynamics. Design/methodology/approach – The numerical technique describing the motion is formulated upon the mixed set of first-order hyperbolic conservation laws already presented by Lee et al. (2013) where the main variables are the linear momentum, the deformation gradient tensor and the total energy. The mixed formulation is discretised using the standard explicit two-step Taylor-Galerkin (2TG) approach, which has been successfully employed in computational fluid dynamics (CFD). Unfortunately, the results display non-physical spurious (or hourglassing) modes, leading to the breakdown of the numerical scheme. For this reason, the 2TG methodology is further improved by means of two ingredients, namely a curl-free projection of the deformation gradient tensor and the inclusion of an additional stiffness stabilisation term. Findings – A series of numerical examples are carried out drawing key comparisons between the proposed formulation and some other recently published numerical techniques. Originality/value – Both velocities (or displacements) and stresses display the same rate of convergence, which proves ideal in the case of industrial applications where low-order discretisations tend to be preferred. The enhancements introduced in this paper enable the use of linear triangular (or bilinear quadrilateral) elements in two dimensional nearly incompressible dynamics applications without locking difficulties. In addition, an artificial viscosity term has been added into the formulation to eliminate the appearance of spurious oscillations in the vicinity of sharp spatial gradients induced by shocks.
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Within the group of immersed boundary methods employed for the numerical simulation of fluid-structure interaction problems, the Immersed Structural Potential Method (ISPM) was recently introduced (Gil et al., 2010) [1] in order to overcome some of the shortcomings of existing immersed methodologies. In the ISPM, an incompressible immersed solid is modelled as a deviatoric strain energy functional whose spatial gradient defines a fluid-structure interaction force field in the Navier-Stokes equations used to resolve the underlying incompressible Newtonian viscous fluid. In this paper, two enhancements of the methodology are presented. First, the introduction of a new family of spline-based kernel functions for the transfer of information between both physics. In contrast to classical IBM kernels, these new kernels are shown not to introduce spurious oscillations in the solution. Second, the use of tensorised Gaussian quadrature rules that allow for accurate and efficient numerical integration of the immersed structural potential. A series of numerical examples will be presented in order to demonstrate the capabilities of the enhanced methodology and to draw some key comparisons against other existing immersed methodologies in terms of accuracy, preservation of the incompressibility constraint and computational speed.
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Continuum immersed strategies are widely used these days for the computational simulation of Fluid–Structure Interaction problems. The principal characteristic of such immersed techniques is the representation of the immersed solid via a momentum forcing source in the Navier–Stokes equations. In this paper, the Immersed Finite Element Method (IFEM), introduced by Zhang et al. (2004) [41] for the analysis of deformable solids immersed in an incompressible Newtonian viscous fluid, is further enhanced by means of three new improvements. A first update deals with the modification of the conservation of mass equation in the background fluid in order to account for non-isochoric deformations within the solid phase. A second update deals with the incompressibility constraint for the solid phase in the case of isochoric deformations, where an enhanced evaluation of the deformation gradient tensor is introduced in a multifield Hu-Washizu variational sense in order to overcome locking effects. The third update is focussed on the improvement of the robustness of the overall scheme, by introducing an implicit one-step time integration scheme with enhanced stability properties, in conjunction with a consistent Newton–Raphson linearisation strategy for optimal quadratic convergence. The resulting monolithic methodology is thoroughly studied for a range of Lagrangian and NURBS based shape finite element functions for a series of numerical examples, with the purpose of studying the effect of the spatial semi-discretisation in the solution. Comparisons are also established with the newly developed Immersed Structural Potential Method (ISPM) by Gil et al. (2010) [7] for benchmarking and assessment of the quality of the new formulation.
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A finite volume cell-centered Lagrangian scheme for solving large deformation problems is constructed based on the hypo-elastic model and using the mimetic theory. Rigorous analysis in the context of gas and solid dynamics, and arbitrary polygonal meshes, is presented to demonstrate the ability of cell-centered schemes in mimicking the continuum properties and principles at the discrete level. A new mimetic formulation based gradient evaluation technique and physics-based, frame independent and symmetry preserving slope limiters are proposed. Furthermore, a physically consistent dissipation model is employed which is both robust and inexpensive to implement. The cell-centered scheme along with these additional new features are applied to solve solids undergoing elasto-plastic deformation.
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A new SUPG-stabilized formulation for Lagrangian hydrodynamics of materials satisfying Mie–Grüneisen equation of state is proposed. It allows the use of simplex-type (triangular/tetrahedral) meshes as well as the more commonly used brick-type (quadrilateral/hexahedral) meshes. The proposed method yields a globally conservative formulation, in which equal-order interpolation (P1 or Q1 isoparametric finite elements) is applied to velocities, displacements, and pressure. As a direct consequence, and in contrast to traditional cell-centered multidimensional hydrocode implementations, the proposed formulation allows a natural representation of the pressure gradient on element interiors. The SUPG stabilization involves additional design requirements, specific to the Lagrangian formulation. A discontinuity capturing operator in the form of a Noh-type viscosity with artificial heat flux is used to preserve stability and smoothness of the solution in shock regions. A set of challenging shock hydrodynamics benchmark tests for the Euler equations of gas dynamics in one and two space dimensions is presented. In the two-dimensional case, computations performed on quadrilateral and triangular grids are analyzed and compared. These results indicate that the new formulation is a promising technology for hydrocode applications.
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We derive a new upwind finite difference approximation to systems of nonlinear hyperbolic conservation laws. The scheme has desirable properties for shock calculations. Under fairly general hypotheses we prove that limit solutions satisfy the entropy condition and that discrete steady shocks exist which are unique and sharp. Numerical examples involving the Euler and Lagrange equations of compressible gas dynamics in one and two space dimensions are given.
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This report presents an assessment of a variety of reconstruction schemes on meshes with both quadrilateral and triangular tessellations. The investigations measure the order of accuracy, absolute error and convergence properties associated with each method. Linear reconstruction approaches using both Green-Gauss and least squares gradient estimation are evaluated against a structured MUSCL scheme wherever possible. In addition to examining the influence of polygon degree and reconstruction strategy, results with three limiters are examined and compared against unlimited results when feasible. The methods are applied on quadrilateral, right triangular, and equilateral triangular elements in order to facilitate an examination of the scheme behavior on a variety of element shapes. The numerical test cases include well known internal and external inviscid examples and also a supersonic vortex problem for which there exists a closed form solution to the 2-D compressible Euler equations. Such investigations indicate that the least squares gradient estimation provides significantly more reliable results on poor quality meshes. Furthermore, limiting only the face normal component of the gradient can significantly increase both accuracy and convergence while still preserving the integral cell average, and maintaining monoticity. The first order method performs poorly on stretched triangular meshes, and analysis shows that such meshes result in poorly aligned left and right states for the Riemann problem. The higher average valence of a vertex in the triangular tessellations does not appear to enhance the wave propagation, accuracy, or convergence properties of the method. Unstructured, Upwind, Inviscid, Reconstruction, Limiters, Riemann problems.
Article
We derive a new upwind finite difference approximation to systems of nonlinear hyperbolic conservation laws. The scheme has desirable properties for shock calculations. Under fairly general hypotheses we prove that limit solutions satisfy the entropy condition and that discrete steady shocks exist which are unique and sharp. Numerical examples involving the Euler and Lagrange equations of compressible gas dynamics in one and two space dimensions are given.
Book
Structures Under Crash and Impact: Continuum Mechanics, Discretization and Experimental Characterization examines the testing and modeling of materials and structures under dynamic loading conditions. Readers will find an in-depth analysis of the current mathematical modeling and simulation tools available for a variety of materials, in addition to both the benefits and limitations they pose in industrial design. The models discussed are also available in commercial codes such as LS-DYNA and AUTODYN. Following a logical and well organized structure, this volume uniquely combines experimental procedures with numerical simulation and features examples from issues taken directly from the automotive, aerospace, and defense industries. Materials scientists, structural and design engineers, and physicists with an interest in crash and impact situations will find Structures Under Crash and Impact a valuable reference. © 2008 Springer Science+Business Media, LLC All rights reserved.
Chapter
In this chapter we apply the mathematical tools presented in Chap. 2 to analyse some of the basic properties of the time–dependent Euler equations. As seen in Chap. 1, the Euler equations result from neglecting the effects of viscosity, heat conduction and body forces on a compressible medium. Here we show that these equations are a system of hyperbolic conservations laws and study some of their mathematical properties. In particular, we study those properties that are essential for finding the solution of the Riemann problem in Chap. 4. We analyse the eigenstructure of the equations, that is, we find eigenvalues and eigenvectors; we study properties of the characteristic fields and establish basic relations across rarefactions, contacts and shock waves. It is worth remarking that the process of finding eigenvalues and eigenvectors usually involves a fair amount of algebra as well as some familiarity with basic physical quantities and their relations. For very complex systems of equations finding eigenvalues and eigenvectors may require the use of symbolic manipulators. Useful background reading for this chapter is found in Chaps. 1 and 2.
Book
Designing engineering components that make optimal use of materials requires consideration of the nonlinear characteristics associated with both manufacturing and working environments. The modeling of these characteristics can only be done through numerical formulation and simulation, and this requires an understanding of both the theoretical background and associated computer solution techniques. By presenting both nonlinear continuum analysis and associated finite element techniques under one roof, Bonet and Wood provide, in this edition of this successful text, a complete, clear, and unified treatment of these important subjects. New chapters dealing with hyperelastic plastic behavior are included, and the authors have thoroughly updated the FLagSHyP program, freely accessible at www.flagshyp.com. Worked examples and exercises complete each chapter, making the text an essential resource for postgraduates studying nonlinear continuum mechanics. It is also ideal for those in industry requiring an appreciation of the way in which their computer simulation programs work.
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This paper presents a novel computational formulation for large strain polyconvex elasticity. The formulation, based on the original ideas introduced by Schröder et al. (2011), introduces the deformation gradient (the fibre map), its adjoint (the area map) and its determinant (the volume map) as independent kinematic variables of a convex strain energy function. Compatibility relationships between these variables and the deformed geometry are enforced by means of a multi-field variational principle with additional constraints. This process allows the use of different approximation spaces for each variable. The paper extends the ideas presented in Schröder et al. (2011) by introducing conjugate stresses to these kinematic variables which can be used to define a generalised convex complementary energy function and a corresponding complementary energy principle of the Hellinger-Reissner type, where the new conjugate stresses are primary variables together with the deformed geometry. Both compressible and incompressible or nearly incompressible elastic models are considered. A key element to the developments presented in the paper is the new use of a tensor cross product, presented for the first time by de Boer (1982), page 76, which facilitates the algebra associated with the adjoint of the deformation gradient. For the numerical examples, quadratic interpolation of the displacements, piecewise linear interpolation of strain and stress fields and piecewise constant interpolation of the Jacobian and its stress conjugate are considered for compressible cases. In the case of incompressible materials two formulations are presented. First, continuous quadratic interpolation for the displacement together with piecewise constant interpolation for the pressure and second, linear continuous interpolation for both displacement and pressure stabilised via a Petrov-Galerkin technique.
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A finite element formulation and algorithm for the nonlinear analysis of the large deflection, materially nonlinear response of impulsively loaded shells is presented. A unique feature of this algorithm is the use of a bilinear four node quadrilateral element with single point quadrature and a simple hourglass control which is orthogonal to straining and rigid body modes on an element level. Numerous results are presented for both elastic and elastic-plastic problems with large strains.
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A mixed second order stabilised Petrov–Galerkin finite element framework was recently introduced by the authors (Lee et al., 2014) [46]. The new mixed formulation, written as a system of conservation laws for the linear momentum and the deformation gradient, performs extremely well in bending dominated scenarios (even when linear tetrahedral elements are used) yielding equal order of convergence for displacements and stresses. In this paper, this formulation is further enhanced for nearly and truly incompressible deformations with three key novelties. First, a new conservation law for the Jacobian of the deformation is added into the system providing extra flexibility to the scheme. Second, a variationally consistent Petrov–Galerkin stabilisation methodology is derived. Third, an adapted fractional step method is presented for both incompressible and nearly incompressible materials in the context of nonlinear elastodynamics. For completeness and ease of understanding, these three improvements are presented both in small and large strain regimes, studying the eigen-structure of the resulting systems. A series of numerical examples are presented in order to demonstrate the robustness of the enhanced methodology with respect to the work previously published by the authors. Ó 2014 Elsevier B.V. All rights reserved.
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SUMMARYA novel finite element (FE) formulation with adaptive mesh rezoning for large deformation problems is proposed. The proposed method takes the advantage of the selective smoothed FE method (S-FEM), which has been recently developed as a locking-free FE formulation with strain smoothing technique. We adopt the selective face-based smoothed/node-based smoothed FEM (FS/NS-FEM-T4) and edge-based smoothed/node-based smoothed FEM (ES/NS-FEM-T3) basically but modify them partly so that our method can handle any kind of material constitutive models other than elastic models. We also present an adaptive mesh rezoning method specialized for our S-FEM formulation with material constitutive models in total form. Because of the modification of the selective S-FEMs and specialization of adaptive mesh rezoning, our method is locking-free for severely large deformation problems even with the use of tetrahedral and triangular meshes. The formulation details for static implicit analysis and several examples of analysis of the proposed method are presented in this paper to demonstrate its efficiency. Copyright © 2014 John Wiley & Sons, Ltd.
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A new family of time integration algorithms is presented for solving structural dynamics problems. The new method, denoted as the generalized-alpha method, possesses numerical dissipation that can be controlled by the user. In particular, it is shown that the generalized-alpha method achieves high-frequency dissipation while minimizing unwanted low-frequency dissipation. Comparisons are given of the generalized-alpha method with other numerically dissipative time integration methods, these results highlight the improved performance of the new algorithm. The new algorithm can be easily implemented into programs that already include the Newmark and Hilber-Hughes-Taylor-alpha time integration methods.