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A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics
Abstract
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.
... 16,17 Gil et al. introduced a three-field Petrov-Galerkin scheme coupled with a total variation diminishing Runge-Kutta method. 18,19 This work also utilizes a variational stabilization technique similar to the dynamic variational multiscale stabilization introduced by Scovazzi et al. 20,21 that permits the usage of linear finite elements for all relevant finite element spaces. Kadapa introduced a semi-implicit scheme based on an explicit form of the Newmark-β method. ...
... By the Courant-Friedrichs-Lewy condition, 39 these wave speeds are crucial for setting constraints on time-step sizes for conditionally stable time integration schemes. 14,18 Because employing the Liu model for the volumetric energy produces a constant instantaneous bulk modulus, this work will use it implicitly when calculating wave speeds and consequent time step size restrictions. J ≈ 1 for incompressible simulations, so these approximations will hold for the quadratic volumetric energy. ...
... As shown in Figure 2, the maximum time step size that guarantees stability for FEBDF2 scales like h/c µ , which is the expected time step size restriction for a semi-implicit method. 15,16,18 For simulations in this work, FEBDF2 was run with a CFL number of 0.5, which was sufficient to guarantee stability for nonlinear problems. 39 ...
The choice of numerical integrator in approximating solutions to dynamic partial differential equations depends on the smallest time-scale of the problem at hand. Large-scale deformations in elastic solids contain both shear waves and bulk waves, the latter of which can travel infinitely fast in incompressible materials. Explicit schemes, which are favored for their efficiency in resolving low-speed dynamics, are bound by time step size restrictions that inversely scale with the fastest wave speed. Implicit schemes can enable larger time step sizes regardless of the wave speeds present, though they are much more computationally expensive. Semi-implicit methods, which are more stable than explicit methods and more efficient than implicit methods, are emerging in the literature, though their applicability to nonlinear elasticity is not extensively studied. In this research, we develop and investigate the functionality of two time integration schemes for the resolution of large-scale dynamics in nearly- and fully-incompressible hyperelastic solids: a Modified Semi-implicit Backward Differentiation Formula integrator (MSBDF2) and a forward Euler / Semi-implicit Backward Differentiation Formula Runge-Kutta integrator (FEBDF2). We prove and empirically verify second order accuracy for both schemes. The stability properties of both methods are derived and numerically verified. We find FEBDF2 has a maximum time step size that inversely scales with the shear wave speed and is unaffected by the bulk wave speed -- the desired stability property of a semi-implicit scheme. Finally, we empirically determine that semi-implicit schemes struggle to preserve volume globally when using nonlinear incompressibility conditions, even under temporal and spatial refinement.
... The authors designed a Petrov-Galerkin finite element formulation for such system to provide stabilization for nearly incompressible materials. In [36], J, the determinant of the deformation gradient, was added as one additional independent variable to enhance the performance in the incompressible limit. In [11], the authors introduced a computational framework for polyconvex hyperelasticity by treating the deformation gradient, its cofactor, and its determinant as independent kinematic variables. ...
... More importantly, considering several different mixed formulations have been proposed in the aforementioned works, we feel the role of those equations needs to be elucidated. In our opinion, the pressure rate equation [102,107,132] or the equation for J [10,36] is a differential mass equation. Consequently, one should judiciously use the classical relation ρJ = ρ 0 , and, in our opinion, it is redundant to solve ρJ = ρ 0 together with the differential mass equation in the discrete problem [107,132]. ...
... A trade-off is that n 2 d additional differential equations for F need to be solved. Based on the polyconvexity hypothesis, it seems natural to introduce kinematic relations for F , JF −T , and J [10,11,36]. In that approach, the kinematic equations involve 2n 2 d + 1 degrees of freedom. ...
We develop a unified continuum modeling framework for viscous fluids and hyperelastic solids using the Gibbs free energy as the thermodynamic potential. This framework naturally leads to a pressure primitive variable formulation for the continuum body, which is well-behaved in both compressible and incompressible regimes. Our derivation also provides a rational justification of the isochoric-volumetric additive split of free energies in nonlinear continuum mechanics. The variational multiscale analysis is performed for the continuum model to construct a foundation for numerical discretization. We first consider the continuum body instantiated as a hyperelastic material and develop a variational multiscale formulation for the hyper-elastodynamic problem. The generalized-alpha method is applied for temporal discretization. A segregated algorithm for the nonlinear solver is designed and carefully analyzed. Second, we apply the new formulation to construct a novel unified formulation for fluid-solid coupled problems. The variational multiscale formulation is utilized for spatial discretization in both fluid and solid subdomains. The generalized-alpha method is applied for the whole continuum body, and optimal high-frequency dissipation is achieved in both fluid and solid subproblems. A new predictor multi-corrector algorithm is developed based on the segregated algorithm to attain a good balance between robustness and efficiency. The efficacy of the new formulations is examined in several benchmark problems. The results indicate that the proposed modeling and numerical methodologies constitute a promising technology for biomedical and engineering applications, particularly those necessitating incompressible models.
... 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. ...
... Expression (1a) represents the standard linear momentum conservation equation, whilst the rest of the Eqs. (1b)-(1d) represent a supplementary set of geometric conservation laws for {F, H, J }. Additionally, appropriate involutions [20,39] must be satisfied by some of the strain variables {F, H} [56] of the system as CURLF = 0; DIVH = 0. ...
... This section introduces the stabilised variational statement for the set of conservation laws (1) via the use of suitable work conjugates [60]. To achieve this, and following References [39,42,43] (13)) alongside the involution Eq. (2). These considerations lead to the expressions for the stabilised conjugate velocity and stresses as described by ...
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.
... However, in case of extreme deformations in the incompressible limit, the p − F formulation lacks robustness. In [46], in the context of the Finite Element Method (FEM), Gil et al. enhanced the formulation for nearly and truly incompressible deformations with the novelty of introducing a conservation law for the Jacobian J of the deformation gradient, providing extra flexibility to the scheme. In [47], a new geometric conservation law for the co-factor H of the deformation gradient was also added to the framework, leading to an enhanced mixed-formulation. ...
... with the shear modulus µ and the Lamé first parameter λ given in terms of the Young's modulus E and Poisson's ratio ν as µ = E 2(1+ν) and λ = Eν (1+ν)(1−2ν) , respectively. Then, based on the derivations presented in [46], a nonlinear approximation is obtained dividing the linear wave speeds (5.41) by the average of the minimum stretch of particles a and b, that is ...
... Conservation of mass, linear and angular momentum are classic in mechanics. Following [44][45][46][47], conservation laws of geometric quantities such as the deformation gradient, the volume map (Jacobian of F ) and the area map (co-factor of F ) are now introduced aiming at the establishment of a mixed set of conservation laws. ...
This work presents a new updated reference Lagrangian Smooth Particle Hydro-dynamics algorithm for the analysis of large deformation by introducing a novel system of first order conservation laws. Both isothermal and thermally-coupled scenarios are considered within the elasticity and elasto-plasticity domains. 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 de-formation gradient, its cofactor and its determinant) are rewritten leading to a very generic (incremental) system of first order conservation laws, which can be degenerated into a total Lagrangian system or into a purely updated Lagrangian system. 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 incremental updated conservation variables which are evolved in time. Taking advantage of this new updated reference Lagrangian formalism, a second order (in space and time) entropy-stable upwiding stabilisation method derived by means of the use of the Rankine Hugoniot jump conditions is introduced. No ad-hoc algorithmic regularisation procedures are needed. To demonstrate the robustness and applicability of the methodology, a wide spectrum of challenging problems are presented and compared, including benchmarks in hyperelasticity, elasto-plasticity and dynamic fracture problems. A new dynamic fracture approach is proposed in this work. The spark for fracture is based on the maximum principal stresses. Once fracture takes place, the particle is split into two new particles and post-fracture velocities and deformation gradients are computed locally, ensuring conservation of mass, linear momentum and total energy. The work explores the use of a series of novel expressions for the evaluation of kernels and the gradients of kernels, all leading to equally robust results and circumventing the issues faced by classic isotropic (spherical) kernels in the presence of strong anisotropic changes in volume.
... 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. Moreover, the {p, F , J} formulation was also extended to account for truly incompressible materials utilising a tailor-made fractional step approach [16,120]. ...
... In order to rectify the shortcomings of classical SPH, Lee et al. [4,5] recently introduced a mixed-based Total Lagrangian SPH computational framework for explicit fast solid dynamics, where the conservation of linear momentum p is solved along with conservation equations for the deformation gradient F , its co-factor H and its Jacobian J. Specifically, the SPH discretisation of the new system of conservation laws {p, F , H, J} was introduced through a family of well-established stabilisation strategies, namely, a Jameson-Schmidt-Turkel (JST) algorithm [3,23] and a variationally consistent Streamline Upwind Petrov Galerkin (SUPG) algorithm [5,17]. Both computational methodologies yield in the same order of convergence is obtained for velocities, deviatoric and volumetric components of stress and were capable of eliminating spurious hourglass-like modes, tensile instability and spurious pressure oscillations in nearly incompressible scenarios. ...
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.
... An alternative way to Eq. (1b) to compute the Jacobian of the deformation is possible via an integral conservation law [29][30][31]80,[84][85][86] as follows, ...
... 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 ...
... In particular, the term D v ab in (29a) addresses spurious zero-energy (hourglass-like [33]) modes due to rank-deficiency, whereas the term D p ab (29b) removes pressure instabilities in near incompressibility regimes [80]. Another challenging aspect when designing a SPH numerical scheme is the ability to control errors arising from the violation of the skew-symmetric nature due to the gradient correction. ...
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. Further enhancement of this framework has been reported by the authors [1,2], when considering isothermal materials governed by a polyconvex constitutive law where the co-factor H of the deformation plays a dominant role. ...
... Using the volumetric energy representation in (35), the pressure p(J, η) can be written as ...
... where F x = ∇ 0 φ(X, t) and ζ F , ζ H , ζ J are dimensionless stabilisation parameters usually in the range of [0, 0.5] [1,2,31,35,84]. ...
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.
... 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 ...
... Finally, the remaining terms to be defined in equations (29a,b) are the so-called pairwise stabilisation terms {D ab v , D ab p , D ab ∆C }. In particular, the term D v ab in (29a) addresses spurious zero-energy (hourglass-like [33]) modes due to rank-deficiency, whereas the term D p ab (29b) removes pressure instabilities in near incompressibility regimes [80]. Another challenging aspect when designing a SPH numerical scheme is the ability to control errors arising from the violation of the skew-symmetric nature due to the gradient correction. ...
... Figure 28 illustatres the deformation history of the water column, displaying pressure contour. As already reported in [47,54,80], pressure instabilities would be excited without incorporating any numerical damping to the Jacobian evolution (see Figure 28a,b). This indeed can be eliminated by introducing a consistent Riemann-based dissipation term D ab p (refer to Figure 28c), without affecting the order of convergence of the algorithm. ...
... 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. ...
... Mixed conservation laws have already been proven [1,21,25] to achieve second order of accuracy for stresses and strains. The two-field mixed formulation, composed of linear momentum p and deformation gradient F, was later augmented by including a governing equation to conserve the Jacobian J of the deformation gradient [18]. This enabled the scheme to effectively solve nearly and fully incompressible materials. ...
... In the place of the full set of conservation variables, denoted here by U = { p, F, H, J }, reduced systems based on { p, F, J }, or only { p, F} formulations have been adopted in the past. The robustness of these reduced systems has been positively ascertained by testing them against a thorough variety of numerical regimes and external conditions [1,18,24]. However, the complete { p, F, H, J } mixed system of conservation laws (2.1) can be provided with mathematicalinstead of merely empirical-proof of existence and stability of physically relevant solutions. ...
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.
... It is known that in the mixed formulation, the Ladyzhenskaya-Babuška-Brezzi (LBB) conditions [1] must be satisfied to obtain stable solutions without spurious oscillations. Several stabilization methods have been proposed to ensure the stability of the mixed formulation, including the use of Taylor-Hood elements [4,32,41], the polynomial pressure projection (PPP) method [20,37], pressure stabilized Petrov--Galerkin formulation (PSPG) [66,26], and variational multiscale (VMS) method [36,10]. However, in the MPM, it is preferable to use basis functions that are at least C 1 continuous to avoid cell-crossing error. ...
... In the final example, we evaluate the performance of the proposed method for a three-dimensional problem involving large deformations. The analysis target is a column subjected to quasi-static torsional deformation, which has been dealt with in several previous studies [26,74,51,50, ...
To further improve the stability of the extended B-spline (EBS) based material point method (EBSMPM), a displacement-pressure (u-p)-mixed formulation incorporating the variational multiscale (VMS) method is formulated within the finite strain framework. In this formulation, we define a constraint on the pressure field that can be applied to both compressible and nearly incompressible materials, taking into account the extensibility to material models that exhibit plastic incompressibility. In order to satisfy the LBB conditions for ensuring stability in the mixed formulation with equal-order interpolation, the VMS method is formulated within finite strain framework in computational solid mechanics. The adoption of the VMS method makes it possible to use higher-order EBS basis functions of the same order for interpolation of both displacement and pressure, and this successfully prevents the occurrence of numerical instability. Three representative numerical examples are presented to demonstrate the performance of the proposed method for solid mechanics problems using compressible and nearly incompressible materials, comparing the formulation whose independent variable is only the displacement field. Additionally, through these numerical examples, the ability to suppress volumetric locking and pressure oscillation is also verified.
... 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]. ...
... However, in case of extreme deformations in the incompressible limit, the {p, F } formulation lacks robustness. In [49], in the context of Finite Element Method (FEM), Gil et al. enhanced the formulation for nearly and truly incompressible deformations with the novelty of introducing a conservation law for the Jacobian J of the deformation gradient, providing extra flexibility to the scheme. In [50], a new geometric conservation law for the co-factor H of the deformation gradient was also added to the framework, leading to an enhanced mixed-formulation, very amenable for constitutive laws strongly dependent on the cofactor of the deformation. ...
... For the same level of accuracy, and due to the higher numerical dissipation of the SPH method, the results obtained with the SPH algorithm have been obtained with slightly finer discretisation than those of the robust Petrov-Galerkin Finite Element Method explored by the authors in their previous publications[48,50,58,59,49]. The latter however requires a few user-defined stabilisation parameters. ...
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.
... Following Refs. [21,22,29], the example of bending column reported in Section 4.2 can be extended to a more challenging test to assess the robustness of TL-SPH-KV for predicting the extremely highly nonlinear deformations. As shown in Fig. 6, the twisting column is also clamped on its bottom face and the body is initialized with a sinusoidal rotational velocity field relative to the origin given by Figure 9 shows the convergence study with particle refinement. ...
... Clearly, TL-SPH produces non-physical stress fluctuations due to insufficient numerical stabilization and then fails to capture the correct deformation pattern. On the contrary, TL-SPH-KV alleviates these discrepancies and predicts more accurate deformation patterns as those in the literature[21,22,29] (see Figs 23 in Ref.[22]), demonstrating the robustness of the proposed method. Note that the results of Ref.[22] (see theirFig. ...
In this paper, we present a simple artificial damping method to enhance the robustness of total Lagrangian smoothed particle hydrodynamics (TL-SPH). Specifically, an artificial damping stress based on the Kelvin-Voigt type damper with a scaling factor imitating a von Neumann-Richtmyer type artificial viscosity is introduced in the constitutive equation to alleviate the spurious oscillation in the vicinity of the sharp spatial gradients. After validating the robustness and accuracy of the present method with a set of benchmark tests with very challenging cases, we demonstrate its potentials in the field of bio-mechanics by simulating the deformation of complex stent structures.
... For truly incompressible material models, fullyimplicit finite element schemes, for example, Rossi et al. [47], Scovazzi et al. [49,50], Liu and Marsden [36,37], and semi-implicit schemes based on fractional-step or projection methods which were originally developed for fluid problems, see Chorin [10,11], Temam [52], Kim and Moin [33], Guermond and Quartapelle [17], Guermond et al. [16], Lovrić et al. [38] are the only techniques available for modelling truly incompressible solids. In literature, Zienkiewicz et al. [59], Lahiri et al. [34] and Gil et al. [14] extended the fraction-step schemes to computational elastodynamics, and Caforio and Imperiale [7] extended a projection method to incompressible elastodynamics. While the implicit schemes become computationally expensive for large-scale models due to the need for inversion of large-scale sparse matrices during the iterative solution procedure at every time step, semi-implicit schemes offer several computational advantages over fully-implicit schemes due to the substantial reduction in the size of matrix systems to be solved for. ...
... The matrices G x and D x in Eq. (43) are the gradient-displacement and divergence-displacement matrices with respect to the current configuration. Similar to (14), G x for the a th basis function of an element is given as ...
This paper presents a novel semi-implicit scheme for elastodynamics and wave propagation problems in nearly and truly incompressible material models. The proposed methodology is based on the efficient computation of the Schur complement for the mixed displacement-pressure formulation using a lumped mass matrix for the displacement field. By treating the deviatoric stress explicitly and the pressure field implicitly, the critical time step is made to be limited by shear wave speed rather than the bulk wave speed. The convergence of the proposed scheme is demonstrated by computing error norms for the recently-proposed LBB-stable BT2/BT1 element. Using the numerical examples modelled with nearly and truly incompressible Neo-Hookean and Ogden material models , it is demonstrated that the proposed semi-implicit scheme yields significant computational benefits over the fully-explicit and the fully-implicit schemes for finite strain elastodynamics simulations involving incompressible materials. Finally, the applicability of the proposed scheme for wave propagation problems in nearly and truly incompressible material models is illustrated.
... 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. ...
... 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 ...
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.
... Some notable contributions towards finite element schemes for performing the simulations of problems modelled with nearly incompressible and elastoplastic material models using triangular/tetrahedral elements are: fractional-step based projection schemes by Zienkiewicz and collaborators [1]; averaged nodal pressure approach by Bonet et al. [2]; node-based uniform strain elements by Dohrmann et al. [3]; stabilised nodally integrated elements by Puso and Solberg [4]; smoothed finite element method by Liu et al. [5]; F-bar patch for triangular and tetrahedral elements by de Souza Neto et al. [6]; 15-node tetrahedral element with reduced-order integration by Danielson [7]; mean-strain 10-node tetrahedral with energy-sampling stabilisation by Pakravan et al. [8,9]; discontinuous Galerkin methods by Hansbo and Larson [10], Noels and Radovitzky [11] and Nguyen and Peraire [12]; mixed-stabilised formulations for solid mechanics problems by by Franca et al. [13], Maniatty and collaborators [14,15,16], Masud and Xia [17,18], Chiumenti and Cervera group [19,20,21,22] and Scovazzi et al. [23,24,25,26,27]; and schemes based on first-order conservation laws by Bonet, Gil and co-workers [28,29]. ...
... In this example, we assess the performance of the proposed finite element scheme under extremely large deformations by studying the problem of twisting of a column studied in [29]. The geometry of the problem and the finite element meshes used for the analysis are shown in Figure 25. ...
We present a novel unified finite element framework for performing computationally efficient large strain implicit and explicit elastodynamic simulations using triangular and tetrahedral meshes that can be generated using the existing mesh generators. For the development of a unified framework, we use Bézier triangular and tetrahedral elements that are directly amenable for explicit schemes using lumped mass matrices and employ a mixed displacement‐pressure formulation for dealing with the numerical issues arising due to volumetric and shear locking. We demonstrate the accuracy of the proposed scheme by studying several challenging benchmark problems in finite strain elastostatics and nonlinear elastodynamics modelled with nearly incompressible hyperelastic and von Mises elastoplastic material models. We show that Bézier elements, in combination with the mixed formulation, help in developing a simple unified finite element formulation that is accurate, robust, and computationally very efficient for performing a wide variety of challenging nonlinear elastostatic and implicit and explicit elastodynamic simulations.
... In the current work, we extend the ALE formalism described above to include thermal effects by solving an additional total energy conservation equation expressed in terms of entropy density within the reference domain. A standout feature of our ALE formulation is its adaptability to simplify into three different mixed-based sets of conservation equations pertinent to solid dynamics: the Total Lagrangian [27][28][29][30][31], Eulerian [32] and Updated Reference Lagrangian formulations [33]. This adaptability can be seen as a generalisation of the velocity-shifting and particle-shifting frameworks previously established in the context of fluid dynamics. ...
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 two considered simulations are: -Two bouncing balls (Fig. 13a) which represent the most basic 2D simulation problem, which uses the MPM builtin contact algorithm. -A twisted column (Fig. 13b) problem inspired by that introduced by Gil et al. [23]. The test consists of a column which is fixed at the bottom and an angular velocity ω 0 = 2π rad/ms is applied to the top surface. ...
The material point method (MPM) is computationally costly and highly parallelisable. With the plateauing of Moore’s law and recent advances in parallel computing, scientists without formal programming training might face challenges in developing fast scientific codes for their research. Parallel programming is intrinsically different to serial programming and may seem daunting to certain scientists, in particular for GPUs. However, recent developments in GPU application programming interfaces (APIs) have made it easier than ever to port codes to GPU. This paper explains how we ported our modular C++ MPM code to GPU without using low-level hardware APIs like CUDA or OpenCL. We aimed to develop a code that has abstracted parallelism and is therefore hardware agnostic. We first present an investigation of a variety of GPU APIs, comparing ease of use, hardware support and performance in an MPM context. Then, the porting process of to the Kokkos ecosystem is detailed, discussing key design patterns and challenges. Finally, our parallel C++ code running on GPU is shown to be up to 85 times faster than on CPU. Since Kokkos also supports Python and Fortran, the principles presented therein can also be applied to codes written in these languages.
... 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]. ...
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.
... 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. ...
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. ...
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.
... (Ingrid S. Lan), amarsden@stanford.edu (Alison L. Marsden) recent works [5,6,7,8]. In our opinion, the unified concept gives rise to promising opportunities for designing new numerical methodologies. ...
In this work, we present a computational fluid-structure interaction (FSI) study for a healthy patient-specific pulmonary arterial tree using the unified continuum and variational multiscale (VMS) formulation we previously developed. The unified framework is particularly well-suited for FSI, as the fluid and solid sub-problems are addressed in essentially the same manner and can thus be uniformly integrated in time with the generalized-α method. In addition, the VMS formulation provides a mechanism for large-eddy simulation in the fluid sub-problem and pressure stabilization in the solid sub-problem. The FSI problem is solved in a quasi-direct approach, in which the pressure and velocity in the unified continuum body are first solved, and the solid displacement is then obtained via a segregated algorithm and prescribed as a boundary condition for the mesh motion. Results of the pulmonary arterial FSI simulation are presented and compared against those of a rigid wall simulation.
... (Ingrid S. Lan), amarsden@stanford.edu (Alison L. Marsden) recent works [5,6,7,8]. In our opinion, the unified concept gives rise to promising opportunities for designing new numerical methodologies. ...
In this work, we present a computational fluid-structure interaction (FSI) study for a healthy patient-specific pulmonary arterial tree using the unified continuum and variational multiscale (VMS) formulation we previously developed. The unified framework is particularly well-suited for FSI, as the fluid and solid sub-problems are addressed in essentially the same manner and can thus be uniformly integrated in time with the generalized-α method. In addition, the VMS formulation provides a mechanism for large-eddy simulation in the fluid sub-problem and pressure stabilization in the solid sub-problem. The FSI problem is solved in a quasi-direct approach, in which the pressure and velocity in the unified continuum body are first solved, and the solid displacement is then obtained via a segregated algorithm and prescribed as a boundary condition for the mesh motion. Results of the pulmonary arterial FSI simulation are presented and compared against those of a rigid wall simulation.
... 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. ...
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.
... (Ingrid S. Lan), amarsden@stanford.edu (Alison L. Marsden) recent works [5,6,7,8]. In our opinion, the unified concept gives rise to promising opportunities for designing new numerical methodologies. ...
In this work, we present a computational fluid-structure interaction (FSI) study for a healthy patient-specific pulmonary arterial tree using the unified continuum and variational multiscale (VMS) formulation we previously developed. The unified framework is particularly well-suited for FSI, as the fluid and solid sub-problems are addressed in essentially the same manner and can thus be uniformly integrated in time with the generalized- method. In addition, the VMS formulation provides a mechanism for large-eddy simulation in the fluid sub-problem and pressure stabilization in the solid sub-problem. The FSI problem is solved in a quasi-direct approach, in which the pressure and velocity in the unified continuum body are first solved, and the solid displacement is then obtained via a segregated algorithm and prescribed as a boundary condition for the mesh motion. Results of the pulmonary arterial FSI simulation are presented and compared against those of a rigid wall simulation.
... 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. ...
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 .
... Finally, we show the applicability of our stabilization techniques for the transient problem of a twisting column [1,40,71]. The initial configuration of the geometry is depicted in Figure 14. ...
Computational formulations for large strain, polyconvex, nearly incompressible elasticity have been extensively studied, but research on enhancing solution schemes that offer better tradeoffs between accuracy, robustness, and computational efficiency remains to be highly relevant. In this paper, we present two methods to overcome locking phenomena, one based on a displacement-pressure formulation using a stable finite element pairing with bubble functions, and another one using a simple pressure-projection stabilized P1-P1 finite element pair. A key advantage is the versatility of the proposed methods: with minor adjustments they are applicable to all kinds of finite elements and generalize easily to transient dynamics. The proposed methods are compared to and verified with standard benchmarks previously reported in the literature. Benchmark results demonstrate that both approaches provide a robust and computationally efficient way of simulating nearly and fully incompressible materials.
... Finally, we show the applicability of our stabilization techniques for the transient problem of a twisting column [1,40,71]. The initial configuration of the geometry is depicted in Fig. 14. ...
Computational formulations for large strain, polyconvex, nearly incompressible elasticity have been extensively studied, but research on enhancing solution schemes that offer better tradeoffs between accuracy, robustness, and computational efficiency remains to be highly relevant. In this paper, we present two methods to overcome locking phenomena, one based on a displacement-pressure formulation using a stable finite element pairing with bubble functions, and another one using a simple pressure-projection stabilized P1-P1 finite element pair. A key advantage is the versatility of the proposed methods: with minor adjustments they are applicable to all kinds of finite elements and generalize easily to transient dynamics. The proposed methods are compared to and verified with standard benchmarks previously reported in the literature. Benchmark results demonstrate that both approaches provide a robust and computationally efficient way of simulating nearly and fully incompressible materials.
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.
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.
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.
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.
We propose a variational multiscale method stabilization of a linear finite element method for nonlinear poroelasticity. Our approach is suitable for the implicit time integration of poroelastic formulations in which the solid skeleton is anisotropic and incompressible. A detailed numerical methodology is presented for a monolithic formulation that includes both structural dynamics and Darcy flow. Our implementation of this methodology is verified using several hyperelastic and poroelastic benchmark cases, and excellent agreement is obtained with the literature. Grid convergence studies for both anisotropic hyperelastodynamics and poroelastodynamics demonstrate that the method is second-order accurate. The capabilities of our approach are demonstrated using a model of the left ventricle (LV) of the heart derived from human imaging data. Simulations using this model indicate that the anisotropicity of the myocardium has a substantial influence on the pore pressure. Furthermore, the temporal variations of the various components of the pore pressure (hydrostatic pressure and pressure resulting from changes in the volume of the pore fluid) are correlated with the variation of the added mass and dynamics of the LV, with maximum pore pressure being obtained at peak systole. The order of magnitude and the temporal variation of the pore pressure are in good agreement with the literature.
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.
This article presents a vertex‐centered 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 discretization 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.
Introducing artificial damping into the momentum equation to enhance the numerical stability of the smoothed particle hydrodynamics (SPH) method for large strain dynamics has been well established, however, implementing appropriate damping term in the constitutive model as an alternative stabilization strategy in the context of the SPH method is still not investigated. In this paper, we present a simple stabilization procedure by introducing an artificial damping term into the second Piola-Kirchhoff stress to enhance the numerical stability of the SPH method in total Lagrangian formulation. The key idea is to reformulate the constitutive equation by adding a Kelvin-Voigt (KV) type damper with a scaling factor imitating a von Neumann-Richtmyer type artificial viscosity to alleviate the spurious oscillation in the vicinity of sharp spatial gradients. The proposed method is shown to effectively eliminate the appearance of spurious non-physical instabilities and easy to be implemented into the original total Lagrangian SPH formulation. After validating the numerical stability and accuracy of the present method through a set of benchmark tests with very challenging cases, we demonstrate its applications and potentials in the field of biomechanics by simulating the deformation of complex stent structures and the electromechanical excitation-contraction of the realistic left ventricle.
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 numerical modelling of fracture has been an active topic of research for over five decades. Most of the approaches employed rely on the use of the Finite Element Method, which has shown to be an effective and cost-efficient tool for solving many physical phenomena. However, the issue of the spurious dependency of the computed solution with the mesh orientation in cracking problems has raised a great concern since its early reports in the 1980s. This matter has proved to be a major challenge in computational solid mechanics; it affects numerous methods employed to solve the problem, in which the computed crack trajectories are spuriously dependent on the arrangement of the finite element (FE) mesh employed. When performing a structural analysis and, in particular, when computing localized failure, it is fundamental to use a reliable and mesh objective method to be able to trust the results produced by the FE code in terms of the fracture
paths, bearing capacity, collapse mechanism and nonlinear responses.
In this doctoral thesis, the mixed ε/u strain/displacement finite element method is used together with multiple isotropic and orthotropic damage constitutive laws for the numerical modelling of quasi-brittle fracture with mesh objectivity. The independent interpolation of the strains increases the accuracy of the computed solution, guaranteeing the local convergence of the stress and strain fields. This feature is a crucial improvement over the standard FE formulation in solid mechanics where the strains are computed as local derivatives of the displacements and the local convergence of the resulting stresses and strains is not ensured. The enhanced precision provided by the mixed formulation in the area near the crack tip is decisive for obtaining unbiased numerical results with
regard to the orientation of the FE mesh.
The strain-driven format of the mixed formulation enables to readily consider different
constitutive laws defined in a stress-strain structure in the numerical simulations. The thesis includes the study of the effect of the material model employed in the resulting crack trajectories as well as the analysis of the relative performance of several isotropic and orthotropic damage behaviors in mode I, mode II, mode III and mixed mode fracture problems.
In this work specific isotropic and orthotropic damage laws are proposed for the numerical modelling of fracture under cyclic loading, which include tensile and compressive damage, stiffness recovery due to crack closure and reopening, as well as irreversible strains. Also, the capacity of the proposed model in reproducing the structural size effect is examined, which is an essential requirement for models aiming at computing quasi-brittle behavior.
In this thesis, a comprehensive comparison of the mixed FE formulation with other techniques employed for computing fracture, specifically the Extended Finite Element Method (XFEM) and the Phase-field model, is made, revealing the cost-efficiency of the proposed Mixed Finite Element Method for modelling quasi-brittle cracking with mesh objectivity. This allows to perform the analysis of real-scale structures, in 2D and 3D, with enhanced accuracy, demonstrating the applicability of this method in the engineering practice. The validation of the model is performed with an extensive comparison of computed results with existing experimental tests and numerical benchmarks. The capacity of the mixed formulation in reproducing forcedisplacement curves, crack trajectories and collapse mechanisms with enhanced accuracy is demonstrated in detail.
Transient elastography is a medical characterization technology that estimates the stiffness of biological soft tissues. By imaging the transient propagation of shear wave in tissues, one can deduce the shear modulus µ. In the last decade, this technique has been used successfully to study various pathologies, particularly fibrosis and cancers. However, numerous factors such as wave reflection, boundary conditions and pre-stress disturb elastography measurements, and the quality of the mechanical characterization of the tissue can be altered. Moreover, the tissues exhibit more complex mechanical properties, including viscosity, nonlinearity and anisotropy, the characterization of which can improve the diagnostic value of elastography. Simulations of wave propagation by finite element (FE) appear promising since they make it possible to study the influence of intrinsic and extrinsic mechanical parameters on the propagation speeds and thus to allow the identification of complex mechanical properties in the real measurement cases. In this work, we develop a FE model for the propagation of nonlinear waves in soft tissues. The numerical models are validated from elastographic experiments taken from the literature, and then used to evaluate the identifiability of the parameters of a nonlinear model in elastography, \emph{i.e.}, Landau's law. By measuring finite amplitude waves and low amplitude waves in pre-deformed states, a practical and robust method is proposed to identify the nonlinearity of homogeneous tissues using elastography experiment. The problem of the cost of computation is also studied in this work. In fact, the quasi-incompressibility of biological tissues makes the compressional wave speed extremely high, which limits the time step of a simulation formulated in explicit dynamics. To deal with this difficulty, different numerical methods are presented, in which the time step is controlled by the shear wave speed instead of the compressional wave speed. Various numerical examples are tested in the context of dynamic elastography, it has been shown that the methods are precise for these problems and a significant reduction of the CPU time is obtained.
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.
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.
We present a computational approach to solve problems in multiplicative nonlinear viscoelasticity using piecewise linear finite elements on triangular and tetrahedral grids, which are very versatile for simulations in complex geometry. Our strategy is based on (1) formulating the equations of mechanics as a mixed first-order system, in which a rate form of the pressure equation is utilized in place of the standard constitutive relationship, and (2) utilizing the variational multiscale approach, in which the stabilization parameter is scaled with the viscous energy dissipation.
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.
Novel unified finite element schemes for computational solid mechanics based on Bézier elements
This paper presents a novel computational framework for the numerical simulation of the electromechanical response of the myocardium during the cardiac cycle. The paper presents the following main novelties. (1) Two new mixed formulations, tailor-made for active stress and active strain coupling approaches, have been developed and used in conjunction with two different ionic models, namely Bueno-Orovio et al. (2008) and Ten Tusscher et al. (2004). Taking as a reference the mixed formulations introduced by Bonet et al. (2015) in the context of nonlinear elasticity, the proposed formulations include as unknown fields the geometry and the transmembrane potential (and possibly a Lagrange multiplier enforcing weakly the incompressibility constraint) as well as the deformation gradient tensor, its cofactor, its determinant, the gradient of the transmembrane potential and their respective work conjugates. The Finite Element implementation of these formulations is shown in this paper, where a static condensation procedure is presented in order to yield an extremely competitive computational approach. (2) A comprehensive and rigorous study of different ionic models (i.e Bueno-Orovio and Ten Tusscher) and electromechanical activation couplings (i.e active strain and active stress) has been carried out. (3) An analytical and numerical analysis of the possible loss of ellipticity and polyconvexity of one of the most widely used constitutive models in the context of cardiac mechanics is carried out in this paper, putting forward possible polyconvexifications of the existing model. (4) In addition, an invariant representation of Guccione's constitutive model is proposed. Finally, a series of numerical examples are included in order to demonstrate the applicability and robustness of the proposed formulations.
A finite-difference method for solving the time-dependent Navier Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an application to a three-dimensional convection problem is presented.
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.
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.
Several finite element methods for large deformation elastic problems in the nearly incompressible and purely incompressible regimes are considered. In particular, the method ability to accurately capture critical loads for the possible occurrence of bifurcation and limit points, is investigated. By means of a couple of 2D model problems involving a very simple neo-Hookean constitutive law, it is shown that within the framework of displacement/pressure mixed elements, even schemes that are inf-sup stable for linear elasticity may exhibit problems when used in the finite deformation regime. The roots of such troubles are identified, but a general strategy to cure them is still missing. Furthermore, a comparison with displacement-based elements, especially of high order, is presented.
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.
A class of useful difference approximations to the full nonlinear Navier-Stokes equations is analyzed; the convergence of these approximations to the solutions of the corresponding differential equations is established and the rate of convergence is estimated.
A Petrov-Galerkin finite element formulation for first-order hyperbolic
systems is developed generalizing the streamline-upwind approach which
has previously been successfully applied to convection-diffusion and
incompressible Navier-Stokes equations. The algorithm is applied to the
Euler equations in conservation-law form and is shown to be effective in
all cases studied, including ones with discontinuous solutions. Accurate
and crisp representation of shock fronts in transonic problems is
achieved.
This paper presents an extension of the energy momentum conserving algorithm, usually developed for hyperelastic constitutive models, to the hypoelastic constitutive models. For such a material no potential can be defined, and thus the conservation of the energy is ensured only if the elastic work of the deformation can be restored by the scheme. We propose a new expression of internal forces at the element level which is shown to verify this property. We also demonstrate that the work of plastic deformation is positive and consistent with the material model. Finally several numerical applications are presented. Copyright © 2003 John Wiley & Sons, Ltd.
In this paper we present a fractional time-step method for Lagrangian formulations of solid dynamics problems. The method can be interpreted as belonging to the class of variational integrators which are designed to conserve linear and angular momentum of the entire mechanical system exactly. Energy fluctuations are found to be minimal and stay bounded for long durations.In order to handle incompressibility, a mixed formulation in which the pressure appears explicitly is adopted. The velocity update over a time step is split into deviatoric and volumetric components. The deviatoric component is advanced using explicit time marching, whereas the pressure correction for each time step is computed implicitly by solving a Poisson-like equation. Once the pressure is known, the volumetric component of the velocity update is calculated. In contrast with standard explicit schemes, where the time-step size is determined by the speed of the pressure waves, the allowable time step for the proposed scheme is found to depend only on the shear wave speed. This leads to a significant advantage in the case of nearly incompressible materials and permits the solution of truly incompressible problems. Copyright © 2005 John Wiley & Sons, Ltd.
Modern computer simulations make stress analysis easy. As they continue to replace classical mathematical methods of analysis, these software programs require users to have a solid understanding of the fundamental principles on which they are based. Develop Intuitive Ability to Identify and Avoid Physically Meaningless Predictions Applied Mechanics of Solids is a powerful tool for understanding how to take advantage of these revolutionary computer advances in the field of solid mechanics. Beginning with a description of the physical and mathematical laws that govern deformation in solids, the text presents modern constitutive equations, as well as analytical and computational methods of stress analysis and fracture mechanics. It also addresses the nonlinear theory of deformable rods, membranes, plates, and shells, and solutions to important boundary and initial value problems in solid mechanics. The author uses the step-by-step manner of a blackboard lecture to explain problem solving methods, often providing the solution to a problem before its derivation is presented. This format will be useful for practicing engineers and scientists who need a quick review of some aspect of solid mechanics, as well as for instructors and students. Select and Combine Topics Using Self-Contained Modules and Subsections Borrowing from the classical literature on linear elasticity, plasticity, and structural mechanics, this book: • Introduces concepts, analytical techniques, and numerical methods used to analyze deformation, stress, and failure in materials or components • Discusses the use of finite element software for stress analysis • Assesses simple analytical solutions to explain how to set up properly posed boundary and initial-value problems • Provides an understanding of algorithms implemented in software code Complemented by the author's website, which features problem sets and sample code for self study, this book offers a crucial overview of problem solving for solid mechanics. It will help readers make optimal use of commercial finite element programs to achieve the most accurate prediction results possible.
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.
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.
The physical understanding of coupled electro-magneto-mechanics has long been a topic of particular importance for scientists. However, it is only in more recent times that the computational mechanics community has been involved, due to the particularly demanding nature of these coupled problems. In this paper, we extend our previous work (Gil AJ, Ledger PD. A coupled hp finite element scheme for the solution of two-dimensional electrostrictive materials. Int J Numer Methods Eng 2012;91:1158–1183) to make it possible to capture more complex coupled phenomena, namely electrostriction and magnetostriction of elastic solids and incompressible Newtonian viscous fluids. From the formulation standpoint, a total Cauchy stress tensor is introduced combining the effects of the mechanical deformation and the ponderomotive force and, for the case of conservative materials, the weak form is obtained from the stationary points of a suitable enthalpy energy formulation. In order to ensure accuracy of results hp-finite elements are employed. Moreover, for computational efficiency, the scheme is implemented in a monolithic manner via a Newton–Raphson strategy with consistent linearisation. A series of well known numerical examples are presented to demonstrate the influence of the electromagnetic phenomena when fully coupled with fluid and solid fields.
A vertex centred Finite Volume algorithm is presented for the numerical simulation of fast transient dynamics problems involving large deformations. A mixed formulation based upon the use of the linear momentum, the deformation gradient tensor and the total energy as conservation variables is discretised in space using linear triangles and tetrahedra in two-dimensional and three-dimensional computations, respectively. The scheme is implemented using central di�fferences for the evaluation of the interface fluxes in conjunction
with the Jameson-Schmidt-Turkel (JST) arti�cial dissipation term. The discretisation in time is performed by using a Total Variational Diminishing (TVD) two-stage Runge-Kutta time integrator. The JST algorithm is
adapted in order to ensure the preservation of linear and angular momenta. The framework results in a low order computationally e�cient solver for solid dynamics, which proves to be very competitive in nearly incompressible scenarios and bending dominated applications.
This updated and expanded edition of the bestselling textbook provides a comprehensive introduction to the methods and theory of nonlinear finite element analysis. New material provides a concise introduction to some of the cutting-edge methods that have evolved in recent years in the field of nonlinear finite element modeling, and includes the eXtended finite element method (XFEM), multiresolution continuum theory for multiscale microstructures, and dislocation-density-based crystalline plasticity.
Nonlinear Finite Elements for Continua and Structures, Second Edition focuses on the formulation and solution of discrete equations for various classes of problems that are of principal interest in applications to solid and structural mechanics. Topics covered include the discretization by finite elements of continua in one dimension and in multi-dimensions; the formulation of constitutive equations for nonlinear materials and large deformations; procedures for the solution of the discrete equations, including considerations of both numerical and multiscale physical instabilities; and the treatment of structural and contact-impact problems.
SUMMARY As part of the ongoing research within the field of computational analysis for the coupled electro-magneto-mechanical response of smart materials, the problem of linearised electrostriction is revisited and analysed for the first time using the framework of hp-finite elements. The governing equations modelling the physics of the dielectric are suitably modified by introducing a new total Cauchy stress tensor (A. Dorfmann and R.W. Ogden. Nonlinear electroelasticity. Acta Mechanica, 174:167–183, 2005), which includes the electrostrictive effect and a staggered partitioned scheme for the numerical solution of the coupling phenomena. With the purpose of benchmarking numerical results, the problem of an infinite electrostrictive plate with a circular/elliptical dielectric insert is revisited. The presented analytical solution is based on the theoretical framework for two-dimensional electrostriction proposed by Knops (R.J. Knops. Two-dimensional electrostriction. Quarterly Journal of Mechanics and Applied Mathematics, 16:377–388, 1963) and uses classical techniques of complex variable analysis. Our presentation, to the best of our knowledge, provides the first correct closed form expression for the solution to the infinite electrostrictive plate with a circular/elliptical dielectric insert, correcting the errors made in previous presentations of this problem. We use this analytical solution to assess the accuracy, efficiency and robustness of the hp-formulation in the case of nearly incompressible electrostrictive materials. Copyright © 2012 John Wiley & Sons, Ltd.
A stabilised second order finite element methodology is presented for the numerical simulation of a mixed conservation law formulation in fast solid dynamics. The mixed formulation, where the unknowns are linear momentum, deformation gradient and total energy, can be cast in the form of a system of first order hyperbolic equations. The difficulty associated with locking effects commonly encountered in standard pure displacement formulations is addressed by treating the deformation gradient as one of the primary variables. The formulation is first discretised in space by using a stabilised Petrov–Galerkin (PG) methodology derived through the use of variational (work-conjugate) principles. The semi-discretised system of equations is then evolved in time by employing a Total Variation Diminishing Runge–Kutta (TVD-RK) time integrator. The formulation achieves optimal convergence (e.g. second order with linear interpolation) with equal orders in velocity (or displacement) and stresses, in contrast with the displacement-based approach. This paper defines a set of appropriate stabilising parameters suitable for this particular formulation, where the results obtained avoid the appearance of non-physical spurious (zero-energy) modes in the solution over a long term response. We also show that the proposed PG formulation is very similar, and under certain conditions identical, to the well known Two-step Taylor Galerkin (2TG). A series of numerical examples are presented in order to assess the performance of the proposed algorithm. The new formulation is proven to be very efficient in nearly incompressible and bending dominated scenarios.
An improved nodal integration method for nearly incompressible materials is proposed. Although nodal integration methods can avoid displacement type locking with nearly incompressible materials, they often encounter pressure oscillations. A mixed approach to nodal integration is exploited to avoid the oscillation. The method is applied to nodal integration using meshless shape functions and appears to work well in simple two dimensional benchmark tests. The method currently relies on a structured discretization but generalizations are proposed as the basis for future work.
In this paper, we present a novel procedure to improve the stress predictions in static, dynamic and nonlinear analyses of solids. We focus on the use of low-order displacement-based finite elements – 3-node and 4-node elements in two-dimensional (2D) solutions, and 4-node and 8-node elements in 3D solutions – because these elements are computationally efficient provided good stress convergence is obtained. We give a variational basis of the new procedure and compare the scheme, and its performance, with other effective previously proposed stress improvement techniques. We observe that the stresses of the new procedure converge quadratically in 1D and 2D solutions, i.e. with the same order as the displacements, and conclude that the new procedure shows much promise for the analysis of solids, structures and multiphysics problems, to calculate improved stress predictions and to establish error measures.
In the past, a number of attempts have failed to robustly compute highly transient shock hydrodynamics flows on tetrahedral meshes. To a certain degree, this is not a surprise, as prior attempts emphasized enhancing the structure of shock-capturing operators rather than focusing on issues of stability with respect to small, linear perturbations. In this work, a new method is devised to stabilize computations on piecewise-linear tetrahedral finite elements. Spurious linear modes are prevented by means of the variational multiscale approach. The resulting algorithm can be proven stable in the linearized limit of acoustic wave propagation. Starting from this solid base, the approach is generalized to fully nonlinear shock computations, by augmenting the discrete formulation with discontinuity-capturing artificial viscosities. Extensive tests in the case of Lagrangian shock dynamics of ideas gases on triangular and tetrahedral grids confirm the stability and accuracy properties of the method. Incidentally, the same tests also reveal the lack of stability of current compatible/mimetic/staggered discretizations: This is due to the presence of specific unstable modes which are theoretically analyzed and verified in computations.
Finite element formulations for arbitrary hyperelastic strain energy functions that are characterized by a locking-free behavior for incompressible materials, a good bending performance and accurate solutions for coarse meshes need still attention. Therefore, the main goal of this contribution is to provide an improved mixed finite element for quasi-incompressible finite elasticity. Based on the knowledge that the minors of the deformation gradient play a major role for the transformation of infinitesimal line-, area- and volume elements, as well as in the formulation of polyconvex strain energy functions a mixed finite element with different interpolation orders of the terms related to the minors is developed. Due to the formulation it is possible to condensate the mixed element formulation at element level to a pure displacement form. Examples show the performance and robustness of the element.
In this paper, we reduce the nonlinear elastic dynamic system, with a polyconvex stored energy function, to a first order
symmetric hyperbolic system in the form of conservation laws.
A cell vertex finite volume algorithm and an artificial compressibility approach are employed to enable simulation of three-dimensional incompressible unsteady turbulent flow using unstructured tetrahedral meshes. Unsteady flow modeling is accomplished through the use of an implicit dual time stepping scheme, and stabilization of the procedure is achieved by the explicit addition of artificial viscosity in the Jameson-Schmidt-Turkel manner. The Spalart-Allmaras detached eddy simulation model is adopted for turbulent flow simulations. The computational performance is enhanced by the incorporation of multigrid acceleration and by parallelization of the solution algorithm. A number of examples are presented to demonstrate the capabilities of the resulting procedure.
A simple four-node quadrilateral and an eight-node hexahedron for large strain analysis of nearly incompressible solids are proposed. Based on the concept of deviatoric/volumetric split and the replacement of the compatible deformation gradient with an assumed modified counterpart, the formulation developed is applicable to arbitrary material models. The closed form of the corresponding exact tangent stiffnesses, which have a particularly simple structure, is derived. It ensures asymptotically quadratic rates of convergence of the Newton-Raphson scheme employed in the solution of the implicit finite element equilibrium equations. From a practical point of view, the incorporation of the proposed elements into existing codes is straightforward. It requires only small changes in the routines of the standard displacement based 4-node quadrilateral and 8-node brick. A comprehensive set of numerical examples, involving hyperelasticity as well as multiplicative elasto-plasticity, is provided. It illustrates the performance of the proposed elements over a wide range of applications, including strain localisation problems, metal forming simulation and adaptive analysis.
This note compares the Hilbur-Hughes-Taylor alpha -method and the Bossak alpha -method for the numerical integration of the equations of vibration of a structure. Details are given in the references cited. 3 refs.
The objective in this paper is to present the method for the calculation of improved stresses published by Payen and Bathe in [1] for the 4-node three-dimensional tetrahedral element. This element is widely used in engineering practice to obtain, in general, only “guiding” results in the analysis of solids because the element is known to be poor in stress predictions. We show in this paper the potential of this novel approach to significantly enhance the stress predictions with the 4-node tetrahedral element at a relatively low computational cost.
The author gives necessary and sufficient conditions for existence and uniqueness of a class of problems of ″saddle point″ type which are often encountered in applying the method of Lagrangian multipliers. A study of the approximation of such problems by means of ″discrete problems″ (with or without numerical integration) is also given, and sufficient conditions for the convergence and error bounds are obtained.
This papers summarizes two linear tetrahedral FE formulations that have been recently proposed to overcome volumetric locking in nearly incompressible explicit dynamic applications. In particular, the average nodal pressure (ANP) technique described by Bonet and Burton (Communications in Numerical Methods in Engineering 1998; 14:437–449) is briefly reviewed. In addition, the split-based formulation proposed by Zienkiewicz et al. (International Journal for Numerical Methods in Engineering 1998; 43:565–583) is described here in terms of a time integration of the nodal Jacobian. This will make it simple to compare both techniques and will enable a new combined method to be presented. The paper will then discuss the stability constraints that each technique places on the timestep size. A von-Neuman stability analysis on simple 1-D uniform meshes will show that the ANP element permits the use of much larger timesteps than the split based formulations. Finally, numerical examples corroborating in 3-D this analytical conclusions will be presented. Copyright © 2001 John Wiley & Sons, Ltd.
A stabilized node-based uniform strain tetrahedral element is presented and analyzed for finite deformation elasticity. The element is based on linear interpolation of a classical displacement-based tetrahedral element formulation but applies nodal averaging of the deformation gradient to improve mechanical behavior, especially in the regime of near-incompressibility where classical linear tetrahedral elements perform very poorly. This uniform strain approach adopted here exhibits spurious modes as has been previously reported in the literature. We present a new type of stabilization exploiting the circumstance that the instability in the formulation is related to the isochoric strain energy contribution only and we therefore present a stabilization based on an isochoric–volumetric splitting of the stress tensor. We demonstrate that by stabilizing the isochoric energy contributions only, reintroduction of volumetric locking through the stabilization can be avoided. The isochoric–volumetric splitting can be applied for all types of materials with only minor restrictions and leads to a formulation that demonstrates impressive performance in examples provided. Copyright © 2008 John Wiley & Sons, Ltd.
This paper revisits the classical discrete geometric conservation law for the arbitrary Lagrangian–Eulerian conservation equations in a moving domain when the conservation laws are mapped to a fixed reference domain. The discretized form of the equations is formulated by means of a cell vertex finite-volume algorithm with special emphasis on the implications of the discrete time–space algorithm for the preservation of free-stream flows. Under these circumstances, two corrections are introduced in order to guarantee the preservation of a free-stream flow evolving in time. A time correction is required to resolve the integration errors in the Jacobian of the transformation or mapping between fixed referential and current domains. A spatial correction is also needed to account for the inexact surface integration in the faces of the dual mesh. Copyright © 2008 John Wiley & Sons, Ltd.
Node-based uniform strain elements for three-node triangular and four-node tetrahedral meshes are presented. The elements use the linear interpolation functions of the original mesh, but each element is associated with a single node. As a result, a favourable constraint ratio for the volumetric response is obtained for problems in solid mechanics. The uniform strain elements do not require the introduction of additional degrees of freedom and their performance is shown to be significantly better than that of three-node triangular or four-node tetrahedral elements. In addition, nodes inside the boundary of the mesh are observed to exhibit superconvergent behaviour for a set of example problems. Published in 2000 by John Wiley & Sons, Ltd.
A stabilized, nodally integrated linear tetrahedral is formulated and analysed. It is well known that linear tetrahedral elements perform poorly in problems with plasticity, nearly incompressible materials, and acute bending. For a variety of reasons, low-order tetrahedral elements are preferable to quadratic tetrahedral elements; particularly for nonlinear problems. But the severe locking problems of tetrahedrals have forced analysts to employ hexahedral formulations for most nonlinear problems. On the other hand, automatic mesh generation is often not feasible for building many 3D hexahedral meshes. A stabilized, nodally integrated linear tetrahedral is developed and shown to perform very well in problems with plasticity, nearly incompressible materials and acute bending. The formulation is analytically and numerically shown to be stable and optimally convergent for the compressible case provided sufficient smoothness of the exact solution u ∈ C2 ∩ (H1)3. Future work may extend the formulation to the incompressible regime and relax the regularity requirements; nonetheless, the results demonstrate that the method is not susceptible to locking and performs quite well in several standard linear and nonlinear benchmarks. Published in 2006 by John Wiley & Sons, Ltd.
This paper provides an assessment of the average nodal volume methodology originally proposed by Bonet and Burton (Commun. Numer. Meth. Engng. 1998; 14:437–449) for the analysis of finitely strained nearly incompressible solids. An implicit version of the average nodal pressure formulation is derived by re-casting the original concept in terms of average nodal volume change ratio within the framework of the F-bar method proposed by de Souza Neto et al. (Int. J. Solids Struct. 1996; 33: 3277–3296). In this context, a linear triangle for implicit plane strain and axisymmetric analysis of nearly incompressible solids under finite strains is obtained. An exact expression for the corresponding element stiffness matrix is presented. This allows the use of the full Newton–Raphson algorithm, ensuring quadratic rates of asymptotic convergence in the global equilibrium iterations. The performance of the procedure is thoroughly assessed by means of numerical examples. The results show that the nodal averaging technique substantially reduces the volumetric locking tendency of the linear triangle and allows an accurate prediction of deformed shapes and reaction forces in situations of practical interest. However, the formulation is found to produce considerable checkerboard-type hydrostatic pressure fluctuations which poses severe limitations on its range of applicability. Copyright © 2004 John Wiley & Sons, Ltd.
This paper presents a simple linear tetrahedron element that can be used in explicit dynamics applications involving nearly incompressible materials or incompressible materials modelled using a penalty formulation. The element prevents volumetric locking by defining nodal volumes and evaluating average nodal pressures in terms of these volumes. Two well-known examples relating to the impact of elasto–plastic bars are used to demonstrate the ability of the element to model large isochoric strains without locking. © 1998 John Wiley & Sons, Ltd.
A numerical time-integration scheme for the dynamics of non-linear elastic shells is presented that simultaneously and independent of the time-step size inherits exactly the conservation laws of total linear, total angular momentum as well as total energy. The proposed technique generalizes to non-linear shells recent work of the authors on non-linear elastodynamics and is ideally suited for long-term/large-scale simulations. The algorithm is second-order accurate and can be immediately extended with no modification to a fourth-order accurate scheme. The property of exact energy conservation induces a strong notion of non-linear numerical stability which manifests itself in actual simulations. The superior performance of the proposed scheme method relative to conventional time-integrators is demonstrated in numerical simulations exhibiting large strains coupled with a large overall rigid motion. These numerical experiments show that symplectic schemes often regarded as unconditionally stable, such as the mid-point rule, can exhibit a dramatic blow-up in finite time while the present method remains perfectly stable.
A class of ‘assumed strain’ mixed finite element methods for fully non-linear problems in solid mechanics is presented which, when restricted to geometrically linear problems, encompasses the classical method of incompatible modes as a particular case. The method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part, formulated in the context of a three-field variational formulation. The resulting class of mixed methods provides a possible extension to the non-linear regime of well-known incompatible mode formulations. In addition, this class of methods includes non-linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements. The good performance of the proposed methodology is illustrated in a number of simulations including 2-D, 3-D and axisymmetric finite deformation problems in elasticity and elastoplasticity. Remarkably, these methods appear to be specially well suited for problems involving localization of the deformation, as illustrated in several numerical examples.
This paper discusses the Bossak–Newmark algorithm, which is an extension of the well‐known Newmark algorithm ¹ for the numerical integration of the equations of discretized structural dynamics problems. The extra parameter introduced here enables the method (when used on the test equation ẍ = −ω ² x ) to be simultaneously second order, unconditionally stable and with positive artificial damping.
Comparisons are made with another modification of Newmark introduced by Hilber, Hughes and Taylor ² .
We offer an alternate derivation for the symmetric-hyperbolic formulation of the equations of motion for a hyperelastic material with polyconvex stored energy. The derivation makes it clear that the expanded system is equivalent, for weak solu-tions, to the original system. We consider motions with variable as well as constant temperature. In addition, we present equivalent Eulerian equations of motion, which are also symmetric-hyperbolic.
A new finite element procedure is introduced for the analysis of nearly-incompressible media. The approach may be simply implemented by a small change of the standard technique, and is applicable to arbitrary anisotropic and/or nonlinear media. The procedure is shown to specialize to the selective integration and mean-dilatation formulations under appropriate hypothesis.
The treatment of zero energy modes which arise due to one-point integration of first-order isoparametric finite elements is addressed. A method for precisely isolating these modes for arbitrary geometry is developed. Two hourglass control schemes, viscous and elastic, are presented and examined. In addition, a convenient one-point integration scheme which analytically integrates the element volume and uniform strain modes is presented.
Explicit dynamic codes which are used currently for the study of plastic deformations in impact, or with some modification for metal forming, suffer two serious limitations.First, only quadrilateral or hexahedral linear elements can be used thus limiting the possibilities of adaptive refinement and adaptive meshing.Second, even with the use of such elements, special devices such as reduced integration must be introduced to avoid locking and reduce costs. These necessitate complex hour glass control, mending-type procedures.The main difficulties are those due to the need of treatment of (almost) incompressible deformation modes. Recently, similar difficulties have been overcome in the context of fluid dynamics soil dynamics and we show here how the processes introduced there can be adopted effectively to the present problem, thus allowing an almost unrestricted choice of element interpolations. © 1998 John Wiley & Sons, Ltd.
This paper presents a new linear tetrahedral element that overcomes the shortcomings in bending dominated problems of the average nodal pressure element presented in Bonet and Burton (Communications in Numerical Methods in Engineering 1998; 14:437–439) Zienkiewicz et al. (Internatinal Journal for Numerical Methods in Engineering 1998; 43:565–583) and Bonet et al. (Internatinal Journal for Numerical Methods in Engineering 2001; 50(1):119–133). This is achieved by extending some of the ideas proposed by Dohrmann et al. (Internatinal Journal for Numerical Methods in Engineering 2000; 47:1549–1568) to the large strain nonlinear kinematics regime. In essence, a nodal deformation gradient is defined by weighted average of the surrounding element values. The associated stresses and internal forces are then derived by differentiation of the corresponding simplified strain energy term. The resulting element is intended for use in explicit dynamic codes (Goudreau and Hallquist, Computer Methods in Applied Mechanics and Engineering 1982; 33) where the use of quadratic tetrahedral elements can present significant difficulties. Copyright © 2001 John Wiley & Sons, Ltd.
A new family of unconditionally stable one-step methods for the direct integration of the equations of structural dynamics is introduced and is shown to possess improved algorithmic damping properties which can be continuously controlled. The new methods are compared with members of the Newmark family, and the Houbolt and Wilson methods.
Some constitutive and computational aspects of finite deformation plasticity are discussed. Attention is restricted to multiplicative theories of plasticity, in which the deformation gradients are assumed to be decomposable into elastic and plastic terms. It is shown by way of consistent linearization of momentum balance that geometric terms arise which are associated with the motion of the intermediate configuration and which in general render the tangent operator non-symmetric even for associated plastic flow. Both explicit (i.e. no equilibrium iteration) and implicit finite element formulations are considered. An assumed strain formulation is used to accommodate the near-incompressibility associated with fully developed isochoric plastic flow. As an example of explicit integration, the rate tangent modulus method is reviewed in some detail. An implicit scheme is derived for which the consistent tangents, resulting in quadratic convergence of the equilibrium iterations, can be written out in closed form for arbitrary material models. All the geometrical terms associated with the motion of the intermediate configuration and the treatment of incompressibility are given explicitly. Examples of application to void growth and coalescence and to crack tip blunting are developed which illustrate the performance of the implicit method.