Development of a cell centred upwind finite volume algorithm for a new conservation law formulation in structural dynamics.

Over the past few decades, dynamic solid mechanics has become a major field of interest in industrial applications involving crash simulation, impact problems, forging and many others to be named. These problems are typically nonlinear due to large deformations (or geometrical nonlinearity) and nonlinear constitutive relations (or material nonlinearity). For this reason, computer simulations for such problems are of practical importance. In these simulations, the Lagrangian formulation is typically used as it automatically satisfies the mass conservation law. Explicit numerical methods are considered to be efficient in these cases. Most of the numerical methods employed for such simulations are developed from the equation of motion (or momentum balance principle). The use of low-order elements in these numerical methods often exhibits the detrimental locking phenomena in the analysis of nearly incompressible applications, which produces an undesirable effect leading to inaccurate results. Situations of this type are usual in the solid dynamics analysis for rubber materials and metal forming processes. In metal plasticity, the plastic deformation is isochoric (or volume-preserving) whereas, the compressible part is due only to elastic deformation. Recently, a new mixed formulation has been established for explicit Lagrangian transient solid dynamics. This formulation, involving linear momentum, deformation gradient and total energy, results in first order hyperbolic system of equations. Such conservation-law formulation enables stresses to converge at the same rate as velocities and displacements. In addition, it ensures that low order elements can be used without volumetric locking and/or bending difficulty for nearly incompressible applications. The new mixed formulation itself shows a clear advantage over the classical displacement-based formulation, due to its simplicity in incorporating state-of-the-art shock capturing techniques. In this research, a curl-preserving cell centred finite volume computational methodology is presented for solving the first order hyperbolic system of conservation laws on quadrilateral cartesian grids. First, by assuming that the approximation to the unknown variables is constant within each cell. This will lead to discontinuities at cell edges which will motivate the use of a Riemann solver by introducing an upwind bias into the evaluation of the numerical flux function. Unfortunately, the accuracy is severely undermined by an excess of numerical dissipation. In order to alleviate this, it is vital to introduce a linear reconstruction procedure for enhancing the accuracy of the scheme. However the second-order spatial method does not prohibit spurious oscillation in the vicinity of sharp gradients. To circumvent this, a nonlinear slope limiter will then be introduced. It is now possible to evolve the semi-discrete evolutionary system of ordinary equations in time with the aid of the family of explicit Total Variation Diminishing Runge Kutta (TVD-RK) time marching schemes. Moreover, a correction procedure involving minimisation algorithm for conservation of the total angular momentum is presented. To this end, a number of interesting examples will be examined in order to demonstrate the robustness and general capabilities of the proposed approach

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

A vertex centred Jameson–Schmidt–Turkel (JST) finite volume algorithm was recently introduced by the authors (Aguirre et al., 2014 [1]) in the context of fast solid isothermal dynamics. The spatial discretisation scheme was constructed upon a Lagrangian two-field mixed (linear momentum and the deformation gradient) formulation presented as a system of conservation laws [2], [3] and [4]. In this paper, the formulation is further enhanced by introducing a novel upwind vertex centred finite volume algorithm with three key novelties. First, a conservation law for the volume map is incorporated into the existing two-field system to extend the range of applications towards the incompressibility limit (Gil et al., 2014 [5]). Second, the use of a linearised Riemann solver and reconstruction limiters is derived for the stabilisation of the scheme together with an efficient edge-based implementation. Third, the treatment of thermo-mechanical processes through a Mie–Grüneisen equation of state is incorporated in the proposed formulation. For completeness, the study of the eigenvalue structure of the resulting system of conservation laws is carried out to demonstrate hyperbolicity and obtain the correct time step bounds for non-isothermal processes. A series of numerical examples are presented in order to assess the robustness of the proposed methodology. The overall scheme shows excellent behaviour in shock and bending dominated nearly incompressible scenarios without spurious pressure oscillations, yielding second order of convergence for both velocities and stresses

A new computational framework for electro-activation in cardiac mechanics

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 [1] and Ten Tusscher [2]. Taking as a reference the mixed formulations introduced by Bonet et al. [3] 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

An entropy-stable Smooth Particle Hydrodynamics algorithm for large strain thermo-elasticity

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 (Lee et al., 2016; Lee et al., 2017; Lee et al., 2019). 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

A Total Lagrangian upwind Smooth Particle Hydrodynamics algorithm for large strain explicit solid dynamics

In previous work [1, 2], Lee et al. introduced a new Smooth Particle Hydrodynamics (SPH) computational framework for large strain explicit solid dynamics with special emphasis on the treatment of near incompressibility. A first order system of hyperbolic equations was presented expressed in terms of the linear momentum and the minors of the deformation, namely the deformation gradient, its co-factor and its Jacobian. Taking advantage of this representation, the suppression of numerical deficiencies (e.g. spurious pressure, long term instability and/or consistency issues) was addressed through well-established stabilisation procedures. In Reference [1], the adaptation of the very efficient Jameson-Schmidt-Turkel algorithm was presented. Reference [2] introduced an adapted variationally consistent Streamline Upwind Petrov Galerkin methodology. In this paper, we now introduce a third alternative stabilisation strategy, extremely competitive, and which does not require the selection of any user-defined artificial stabilisation parameter. Specifically, a characteristic-based Riemann solver in conjunction with a linear reconstruction procedure is used, with the aim to guarantee both consistency and conservation of the overall algorithm. We show that the proposed SPH formulation is very similar in nature to that of the upwind vertex centred Finite Volume Method presented in [3]. In order to extend the application range towards the incompressibility limit, an artificial compressibility algorithm is also developed. Finally, an extensive set of challenging numerical examples is analysed. The new SPH algorithm shows excellent behaviour in compressible, nearly incompressible and truly incompressible scenarios, yielding second order of convergence for velocities, deviatoric and volumetric components of the stress

A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity

In Parts I [1] and II [2] 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-Gruneisen 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

A new Jameson–Schmidt–Turkel Smooth Particle Hydrodynamics algorithm for large strain explicit fast dynamics

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 (Bonet et al., 2015; Gil et al., 2016) 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 (Jameson et al., 1981) 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

An upwind cell centred Total Lagrangian finite volume algorithm for nearly incompressible explicit fast solid dynamic applications

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 (k / m > 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

A variationally consistent Streamline Upwind Petrov–Galerkin Smooth Particle Hydrodynamics algorithm for large strain solid dynamics

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 Lee et al. (2016) 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 Lee, Gil and Bonet (2014), Gil et al. (2014), Gil et al. (2016), Bonet et al. (2015), 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

A first-order hyperbolic arbitrary Lagrangian Eulerian conservation formulation for non-linear solid dynamics

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