Content uploaded by Chun Hean Lee

Author content

All content in this area was uploaded by Chun Hean Lee on Jun 18, 2018

Content may be subject to copyright.

A preview of the PDF is not available

The paper presents a new computational framework for the numerical simulation of fast large strain solid dynamics, with particular emphasis on the treatment of near incompressibility. A complete set of first order hyperbolic conservation equations expressed in terms of the linear momentum and the minors of the deformation (namely the deformation gradient, its co-factor and its Jacobian), in conjunction with a polyconvex nearly incompressible constitutive law, is presented. Taking advantage of this elegant formalism , alternative implementations in terms of entropy-conjugate variables are also possible, through suitable symmetrisation of the original system of conservation variables. From the spatial discretisation standpoint, modern Computational Fluid Dynamics code "OpenFOAM" [http://www.openfoam.com/] is here adapted to the field of solid mechanics, with the aim to bridge the gap between computational fluid and solid dynamics. A cell centred finite volume algorithm is employed and suitably adapted. Naturally, discontinuity of the conservation variables across control volume interfaces leads to a Riemann problem, whose resolution requires special attention when attempting to model materials with predominant nearly incompressible behaviour (κ/µ ≥ 500). For this reason, an acoustic Riemann solver combined with a preconditioning procedure is introduced. In addition, a global a posteriori angular momentum projection procedure proposed in [1] is also presented and adapted to a Total Lagrangian version of the nodal scheme of Kluth and Després [2] used in this paper for comparison purposes. Finally, a series of challenging numerical examples is examined in order to assess the robustness and applicability of the proposed methodology with an eye on large scale simulation in future works.

Figures - uploaded by Chun Hean Lee

Author content

All figure content in this area was uploaded by Chun Hean Lee

Content may be subject to copyright.

Content uploaded by Chun Hean Lee

Author content

All content in this area was uploaded by Chun Hean Lee on Jun 18, 2018

Content may be subject to copyright.

A preview of the PDF is not available

... For the particular case of a nearly incompressible neo-Hookean material, the multivariable strain energy Ψ X (F , H, J) can be decomposed into the summation of de-viatoricΨ X and volumetric U contributions as [74] ...

... This example was previously studied in an isothermal context in [74] and thermal effects were recently considered in [102]. The model consists of a block 1m x 1m x 0.5m, including nine holes with diameter equal to 0.2m each. ...

... with Q χ = −K χ ∇ χ θ, (C. 74) and ...

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.

... Second, since there are no dissipative or external forces involved, it presents an ideal test for verifying the conservation of energy, which we achieve to a reasonable degree, see Figure 3. Last but not least, due to the presence of strongly negative pressures, this simulation poses a challenge with respect to tensile instability, demonstrating the usefulness of the penalty term (19). In fact, without this addition, the plate would tear completely as can be seen in Figure 4 ...

... The next benchmark is borrowed from [19]. The initial setup is a cuboid ...

... Although the column is shrinking at first, it "bounces off" at some point and starts to elongate. This effect was also observed in [4] and [19]. The blue line shows comparison to a finite element simulation (created using the Fenics software [23]). ...

Smoothed Particle Hydrodynamics (SPH) methods are advantageous in simulations of fluids in domains with free boundary. Special SPH methods have also been developed to simulate solids. However, there are situations where the matter behaves partly as a fluid and partly as a solid, for instance, the solidification front in 3D printing, or any system involving both fluid and solid phases. We develop an SPH-like method that is suitable for both fluids and solids at the same time. Instead of the typical discretization of hydrodynamics, we discretize the Symmetric Hyperbolic Thermodynamically Compatible equations (SHTC), which describe both fluids, elastic solids, and visco-elasto-plastic solids within a single framework. The resulting SHTC-SPH method is then tested on various benchmarks from the hydrodynamics and dynamics of solids and shows remarkable agreement with the data.

... Consider the three dimensional deformation of an isothermal body of material density ρ R moving from its initial undeformed configuration Ω V , with boundary ∂Ω V defined by an outward unit normal N , to a current deformed configuration Ω v (t) at time t, with boundary ∂Ω v (t) defined by an outward unit normal n. The time dependent motion ϕ(X, t) of the body can be described by the following system of Total Lagrangian global conservation laws [35,[39][40][41][42][43][44][45][46][47][48][49][50][51][52] ...

... Numerically, expressions (34) can be viewed as the summation of the average states (unstable) and the associated upwinding stabilisation terms depending on the jumps. This has been extensively exploited by the authors in developing stabilised methods with the objective to improve the numerical solutions by alleviating unwanted spurious hour-glassing and pressure instabilities [34,43,45,[47][48][49]. Provided the interface conditions (25) and also the slip function Φ ≤ 0 (26) hold, we accept the contact-stick solution (34) as the actual local solution for contact. ...

... Consider the case of elasto-plasticity [47,52,58] where the elastic energy is expressed in terms of elastic left Cauchy-Green tensor b e = F C −1 p F T , thus in this case the internal state variable is indeed the inverse of the plastic right Cauchy Green tensor, that is α = C −1 p . With this, the rate of plastic dissipationḊ described in (54) becomeṡ ...

This paper presents a vertex‐centred finite volume algorithm for the explicit dynamic analysis of large strain contact problems. The methodology exploits the use of a system of first order conservation equations written in terms of the linear momentum and a triplet of geometric deformation measures (comprising the deformation gradient tensor, its co‐factor and its Jacobian) together with their associated jump conditions. The latter can be used to derive several dynamic contact models ensuring the preservation of hyperbolic characteristic structure across solution discontinuities at the contact interface, a clear advantage over the standard quasi‐static contact models where the influence of inertial effects at the contact interface is completely neglected. Taking advantage of the conservative nature of the formalism, both kinetic (traction) and kinematic (velocity) contact interface conditions are explicitly enforced at the fluxes through the use of appropriate jump conditions. Specifically, the kinetic condition is enforced in the usual linear momentum equation, whereas the kinematic condition can now be easily enforced in the geometric conservation equations without requiring a computationally demanding iterative algorithm. Additionally, a Total Variation Diminishing shock capturing technique can be suitably incorporated in order to improve dramatically the performance of the algorithm at the vicinity of shocks. Moreover, and to guarantee stability from the spatial discretisation standpoint, global entropy production is demonstrated through the satisfaction of semi‐discrete version of the classical Colemanâ€“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.

... Second, since there are no dissipative or external forces involved, it presents an ideal test for verifying the conservation of energy, which we achieve to a reasonable degree, see Figure 3. Last but not least, due to the presence of strongly negative pressures, this simulation poses a challenge with respect to tensile instability, demonstrating the usefulness of the penalty term (19). In fact, without this addition, the plate would tear completely as can be seen in Figure 4 ...

... The next benchmark is borrowed from [19]. The initial setup is a cuboid ...

... Although the column is shrinking at first, it "bounces off" at some point and starts to elongate. This effect was also observed in [4] and [19]. The blue line shows comparison to a finite element simulation (created using the Fenics software [23]). ...

Smoothed Particle Hydrodynamics (SPH) methods are advantageous in simulations of fluids in domains with free boundary. Special SPH methods have also been developed to simulate solids. However, there are situations where the matter behaves partly as a fluid and partly as a solid, for instance, the solidification front in 3D printing, or any system involving both fluid and solid phases. We develop an SPH-like method that is suitable for both fluids and solids at the same time. Instead of the typical discretization of hydrodynamics, we discretize the Symmetric Hyperbolic Thermodynamically Compatible equations (SHTC), which describe both fluids, elastic solids, and visco-elasto-plastic solids within a single framework. The resulting SHTC-SPH method is then tested on various benchmarks from the hydrodynamics and dynamics of solids and shows remarkable agreement with the data.

... 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]. ...

... It should be mentioned that there are other finite volume approaches in which the development of contact modelling is lately gaining more attention. An example are works from Bessenov et al., 20 Rucnie et al., 21 and Haider et al. 22 in which the authors propose contact algorithms intended for explicit finite difference and vertex-centred finite volume frameworks. ...

This article presents a new implicit coupling procedure for mechanical contact simulations using an implicit cell-centred finite volume method. Both contact boundaries are treated as Neumann conditions, where the prescribed contact force is calculated using a penalty law, which is linearised and updated within the iterative solution procedure. Compared to the currently available explicit treatment, the implicit treatment offers better efficiency for the same accuracy. This is achieved with the proposed implicit linearisation, which replaces the explicit under-relaxation of the contact force. The proposed procedure, intended for frictionless contact of Hookean solids, can handle non-conformal contact interface discretisations and faces in partial contact. The accuracy and efficiency of the implicit approach are compared with the explicit procedure on four benchmark problems, where it is shown that the proposed method can significantly improve efficiency and robustness.

... 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.

... Consider the three dimensional deformation of an isothermal body of material density ρ R moving from its initial undeformed configuration Ω V , with boundary ∂Ω V defined by an outward unit normal N , to a current deformed configuration Ω v at time t, with boundary ∂Ω v defined by an outward unit normal n (see Figure 1). The time dependent motion ϕ(X, t) of the body can be described by a system of first order Total Lagrangian conservation laws [1,2,3,54,55], summarised as ...

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.

This paper presents a high-order finite volume method for solving linear elasticity problems on two-dimensional unstructured meshes. The method is designed to increase the effectiveness of finite volume methods in solving structural problems affected by shear locking.
The particular feature of the proposed method is the use of Moving Least Squares (MLS) and Local Regression Estimators (LRE). Unlike other approaches proposed before, these interpolation schemes lead to a natural and simple extension of the classical finite volume method to arbitrary order. The unknowns of the problem are still the nodal values of the displacement which are obtained implicitly in a direct solution strategy.
Some canonical tests are performed to demonstrate the accuracy of the method. An analytical example is considered to evaluate the sensitivity of the solution concerning the parameters of the algorithm. A thin curved beam and a crack problem are considered to show that the method can deal with the shear locking effect, stress concentrations, and geometries where unstructured meshes are required. An overall better behavior of the LRE is observed. A comparison between low and high-order schemes is presented, and a set of parameters for the interpolation method is found, delivering good results for the proposed cases.

A cell-centered implicit-explicit updated Lagrangian finite volume scheme on unstructured grids is proposed for a unified first-order hyperbolic formulation of continuum fluid and solid mechanics, namely the Godunov-Peshkov-Romenski (GPR) model. The scheme provably respects the stiff relaxation limits of the continuous model at the fully discrete level, thus it is asymptotic preserving. Furthermore, the GCL is satisfied by a compatible discretization that makes use of a nodal solver to compute vertex-based fluxes that are used both for the motion of the computational mesh as well as for the time evolution of the governing PDEs. Second order of accuracy in space is achieved using a TVD piecewise linear reconstruction, while an implicit-explicit (IMEX) Runge-Kutta time discretization allows the scheme to obtain higher accuracy also in time. Particular care is devoted to the design of a stiff ODE solver, based on approximate analytical solutions of the governing equations, that plays a crucial role when the visco-plastic limit of the model is approached. We demonstrate the accuracy and robustness of the scheme on a wide spectrum of material responses covered by the unified continuum model that includes inviscid hydrodynamics, viscous heat conducting fluids, elastic and elasto-plastic solids in multidimensional settings.

Designing engineering components that make optimal use of materials requires consideration of the nonlinear static and dynamic characteristics associated with both manufacturing and working environments. The modeling of these characteristics can only be done through numerical formulation and simulation, which requires an understanding of both the theoretical background and associated computer solution techniques. By presenting both the nonlinear solid mechanics and the associated finite element techniques together, the authors provide, in the first of two books in this series, a complete, clear, and unified treatment of the static aspects of nonlinear solid mechanics. Alongside a range of worked examples and exercises are user instructions, program descriptions, and examples for the FLagSHyP MATLAB computer implementation, for which the source code is available online. While this book is designed to complement postgraduate courses, it is also relevant to those in industry requiring an appreciation of the way their computer simulation programs work. © Javier Bonet, Antonio J. Gil, and Richard D. Wood 2016. All rights reserved.

A new nodal mixed finite element is proposed for the simulation of linear elastodynamics and wave propagation problems in time domain. Our method is based on equal-order interpolation discrete spaces for both the velocity (or displacement) and stress (or strain) tensor variables. The mixed form is derived using either the velocity/stress or velocity/strain pair of unknowns, the latter being instrumental in extensions of the method to nonlinear mechanics. The proposed approach works equally well on hexahedral or tetrahedral grids and, for this reason, it is suitable for time-domain engineering applications in complex geometry. The peculiarity of the proposed approach is the use of the rate form of the stress update equation, which yields a set of governing equations with the structure of a non-dissipative space/time Friedrichs’ system. We complement standard traction boundary conditions for the stress with strongly and weakly enforced boundary conditions for the velocity (or displacement). Weakly enforced boundary conditions are particularly suitable when considering complex geometrical shapes, because they do not require dedicated data structures for the imposition of the boundary degrees of freedom, but, rather, they utilize the structure of the variational formulation. We also show how the framework of weakly enforced boundary conditions can be used to develop variational forms for multi-domain simulations of heterogenous media. A complete analysis including stability and convergence proofs is included, in the case of a space–time variational appraoch. A series of computational tests are used to demonstrate and verify the performance of the proposed approach.

In this article, we develop a dynamic version of the variational multiscale (D-VMS) stabilization for nearly/fully incompressible solid dynamics simulations of viscoelastic materials. The constitutive models considered here are based on Prony series expansions, which are rather common in the practice of finite element simulations, especially in industrial/commercial applications. Our method is based on a mixed formulation, in which the momentum equation is complemented by a pressure equation in rate form. The unknown pressure, displacement and velocity are approximated with piecewise linear, continuous finite element functions. In order to prevent spurious oscillations, the pressure equation is augmented with a stabilization operator specifically designed for viscoelastic problems, in that it depends on the viscoelastic dissipation. We demonstrate the robustness, stability, and accuracy properties of the proposed method with extensive numerical tests in the case of linear and finite deformations. This article is protected by copyright. All rights reserved.

This paper presents a new Smooth Particle Hydrodynamics (SPH) computational framework for explicit fast solid dynamics. The proposed methodology explores the use of the Streamline Upwind Petrov Galerkin (SUPG) stabilisation methodology as an alternative to the Jameson-Schmidt-Turkel (JST) stabilisation recently presented by the authors in [1] in the context of a conservation law formulation of fast solid dynamics. The work introduced in this paper puts forward three advantageous features over the recent JST-SPH framework. First, the variationally consistent nature of the SUPG stabilisation allows for the introduction of a locally preserving angular momentum procedure which can be solved in a monolithic manner in conjunction with the rest of the system equations. This differs from the JST-SPH framework, where an a posteriori projection procedure was required to ensure global angular momentum preservation. Second, evaluation of expensive harmonic and bi-harmonic operators , necessary for the JST stabilisation, is circumvented in the new SUPG-SPH framework. Third, the SUPG-SPH framework is more accurate (for the same number of degrees of freedom) than its JST-SPH counterpart and its accuracy is comparable to that of the robust (but computationally more demanding) Petrov Galerkin Finite Element Method (PG-FEM) technique explored by the authors in [2–5], as shown in the numerical examples included. A series of numerical examples are analysed in order to benchmark and assess the robustness and effectiveness of the proposed algorithm. The resulting SUPG-SPH framework is therefore accurate, robust and computationally efficient, three key desired features that will allow the authors in forthcoming publications to explore its applicability in large scale simulations.

The Finite Volume discretization of non-linear elasticity equations seems to be a promising alternative to the traditional Finite Element discretization as mentioned by Lee et al. [Computers and Structures (2013)]. In this work, we propose to solve the elastic response of a solid material by using a cell-centered Finite Volume Lagrangian scheme in the current configuration. The hyperelastic approach is chosen for representing elastic isotropic materials. In this way, the constitutive law is based on the principle of frame indifference and thermodynamic consistency which are imposed by mean of the Coleman-Noll procedure. It results in defining the Cauchy stress tensor as the derivative of the free-energy with respect to the left Cauchy-Green tensor. Moreover, the materials being isotropic, the free-energy is function of the left Cauchy-Green tensor invariants which enables the use of the Neo-Hookean model. The hyperelasticity system is discretized using the cell-centered Lagrangian scheme from the work of Maire et al. [J. Comput. Phys. (2009)]. The 3D scheme is first order in space and time and is assessed against three test cases with both infinitesimal displacements and large deformations to show the good accordance between the numerical solutions and the analytic ones. This article is protected by copyright. All rights reserved.

We propose a stabilization method for linear tetrahedral finite elements, suitable for the implicit time integration of the equations of nearly and fully incompressible nonlinear elastodynamics. In particular, we derive and discuss a generalized framework for stabilization and implicit time integration that can comprehensively be applied to the class of all isotropic hyperelastic models. In this sense the presented development can be considered an important extension and complement to the stabilization approach proposed by the authors in previous work, which was instead focused on explicit time integration and simple neo-Hookean models for nearly-incompressible elasticity. With the goal of computational efficiency, we also present a two-step block Gauss–Seidel strategy for the time update of displacements, velocities and pressures. Specifically, a mixed system of equations for the velocity and pressure is updated implicitly in a first stage, and the displacements are updated explicitly in a second stage. The proposed mixed formulation is then embedded in Newton-type strategies for the nonlinear solution of the equations of motion. Various implicit time integration strategies are considered, and, particularly, we focus on high-frequency dissipation time integrators, which are preferable in transient mechanics applications. An extensive set of numerical computations with linear tetrahedral elements is presented to demonstrate the performance of the proposed approach.

The current article presents a Lagrangian cell-centred finite volume solution methodology for simulation of metal forming processes. Details are given of the mathematical model in updated Lagrangian form, where a hyperelastoplastic J2 constitutive relation has been employed. The cell-centred finite volume discretisation is described, where a modified discretised is proposed to alleviate erroneous hydrostatic pressure oscillations; an outline of the memory efficient segregated solution procedure is given. The accuracy and order of accuracy of the method is examined on a number of 2-D and 3-D elastoplastic benchmark test cases, where good agreement with available analytical and finite element solutions is achieved. This article is protected by copyright. All rights reserved.

This paper presents a new Smooth Particle Hydrodynamics (SPH) computational framework for large strain explicit solid dynamics. A mixed-based set of Total Lagrangian conservation laws is presented in terms of the linear momentum and an extended set of geometric strain measures, comprised of the deformation gradient, its co-factor and the Jacobian. Taking advantage of this representation, the main aim of this paper is the adaptation of the very efficient Jameson-Schmidt-Turkel (JST) algorithm, extensively used in computational fluid dynamics, to a SPH based discretisation of the mixed-based set of conservation laws, with three key distinct novelties. First, a conservative JST-based SPH computational framework is presented with emphasis in nearly incompressible materials. Second, the suppression of numerical instabilities associated with the non-physical zero-energy modes is addressed through a well-established stabilisation procedure. Third, the use of a discrete angular momentum projection algorithm is presented in conjunction with a monolithic Total Variation Diminishing Runge-Kutta time integrator in order to guarantee the global conservation of angular momentum. For completeness, exact enforcement of essential boundary conditions is incorporated through the use of a Lagrange multiplier projection technique. A series of challenging numerical examples (e.g. in the near incompressibility regime) are examined in order to assess the robustness and accuracy of the proposed algorithm. The obtained results are benchmarked against a wide spectrum of alternative numerical strategies.

We analyze one-sided or upwind finite difference approximations to hyperbolic partial differential equations and, in particular, nonlinear conservation laws. Second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions. We show that no such stable approximation of order higher than two is possible. These one-sided schemes have desirable properties for shock calculations. We show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance. New and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles, existence, uniqueness, and stability of discrete shocks. Numerical examples are given.