A structure-preserving integrator for incompressible finite elastodynamics based on a grad-div stabilized mixed formulation with particular emphasis on stretch-based material models

We present a structure-preserving scheme based on a recently-proposed mixed formulation for incompressible hyperelasticity formulated in principal stretches. Although there exist Hamiltonians introduced for quasi-incompressible elastodynamics based on different variational formulations, the one in the fully incompressible regime has yet been identified in the literature. The adopted mixed formulation naturally provides a new Hamiltonian for fully incompressible elastodynamics. Invoking the discrete gradient formula, we are able to design fully-discrete schemes that preserve the Hamiltonian and momenta. The scaled mid-point formula, another popular option for constructing algorithmic stresses, is analyzed and demonstrated to be non-robust numerically. The generalized Taylor-Hood element based on the spline technology conveniently provides a higher-order, robust, and inf-sup stable spatial discretization option for finite strain analysis. To enhance the element performance in volume conservation, the grad-div stabilization, a technique initially developed in computational fluid dynamics, is introduced here for elastodynamics. It is shown that the stabilization term does not impose additional restrictions for the algorithmic stress to respect the invariants, leading to an energy-decaying and momentum-conserving fully discrete scheme. A set of numerical examples is provided to justify the claimed properties. The grad-div stabilization is found to enhance the discrete mass conservation effectively. Furthermore, in contrast to conventional algorithms based on Cardano's formula and perturbation techniques, the spectral decomposition algorithm developed by Scherzinger and Dohrmann is robust and accurate to ensure the discrete conservation laws and is thus recommended for stretch-based material modeling

A dissipative discretization for large deformation frictionless dynamic contact problems

We present a discretization for dynamic large deformation contact problems without friction. Our model is based on Hamilton’s principle, which avoids the explicit appearance of the contact forces. The resulting differential inclusion is discretized in time using a modified midpoint rule. This modification, which concerns the evaluation of the generalized gradient, allows to achieve energy dissipativity. For the space discretization we use a dual-basis mortar method. The resulting spatial algebraic problems are nonconvex minimization problems with nonconvex inequality constraints. These can be solved efficiently using a trust-region SQP framework with a monotone multigrid inner solver

Variational Bonded Discrete Element Method with Manifold Optimization

This paper proposes a novel approach that combines variational integration with the bonded discrete element method (BDEM) to achieve faster and more accurate fracture simulations. The approach leverages the efficiency of implicit integration and the accuracy of BDEM in modeling fracture phenomena. We introduce a variational integrator and a manifold optimization approach utilizing a nullspace operator to speed up the solving of quaternion-constrained systems. Additionally, the paper presents an element packing and surface reconstruction method specifically designed for bonded discrete element methods. Results from the experiments prove that the proposed method offers 2.8 to 12 times faster state-of-the-art methods

Discretisation techniques for large deformation computational contact elastodynamics

The present thesis deals with large deformation contact problems of flexible bodies in the field of nonlinear elastodynamics. Special emphasis will be placed on a consistent spatial and temporal discretization. For the spatial discretization of the underlying bodies, the finite element method will be used. For the contact boundaries the collocation type node-to-surface method as well as the variationally consistent Mortar method will be applied

A novel mixed and energy‐momentum consistent framework for coupled nonlinear thermo‐electro‐elastodynamics

A novel mixed framework and energy-momentum consistent integration scheme in the field of coupled nonlinear thermo-electro-elastodynamics is proposed. The mixed environment is primarily based on a framework for elastodynamics in the case of polyconvex strain energy functions. For this elastodynamic framework, the properties of the so-called tensor cross product are exploited to derive a mixed formulation via a Hu-Washizu type extension of the strain energy function. Afterwards, a general path to incorporate nonpotential problems for mixed formulations is demonstrated. To this end, the strong form of the mixed framework is derived and supplemented with the energy balance as well as Maxwell\u27s equations neglecting magnetic and time dependent effects. By additionally choosing an appropriate energy function, this procedure leads to a fully coupled thermo-electro-elastodynamic formulation which benefits from the properties of the underlying mixed framework. In addition, the proposed mixed framework facilitates the design of a new energy-momentum consistent time integration scheme by employing discrete derivatives in the sense of Gonzalez. A one-step integration scheme of second-order accuracy is obtained which is shown to be stable even for large time steps. Eventually, the performance of the novel formulation is demonstrated in several numerical examples

Fast Nonlinear Least Squares Optimization of Large-Scale Semi-Sparse Problems

Many problems in computer graphics and vision can be formulated as a nonlinear least squares optimization problem, for which numerous off-the-shelf solvers are readily available. Depending on the structure of the problem, however, existing solvers may be more or less suitable, and in some cases the solution comes at the cost of lengthy convergence times. One such case is semi-sparse optimization problems, emerging for example in localized facial performance reconstruction, where the nonlinear least squares problem can be composed of hundreds of thousands of cost functions, each one involving many of the optimization parameters. While such problems can be solved with existing solvers, the computation time can severely hinder the applicability of these methods. We introduce a novel iterative solver for nonlinear least squares optimization of large-scale semi-sparse problems. We use the nonlinear Levenberg-Marquardt method to locally linearize the problem in parallel, based on its firstorder approximation. Then, we decompose the linear problem in small blocks, using the local Schur complement, leading to a more compact linear system without loss of information. The resulting system is dense but its size is small enough to be solved using a parallel direct method in a short amount of time. The main benefit we get by using such an approach is that the overall optimization process is entirely parallel and scalable, making it suitable to be mapped onto graphics hardware (GPU). By using our minimizer, results are obtained up to one order of magnitude faster than other existing solvers, without sacrificing the generality and the accuracy of the model. We provide a detailed analysis of our approach and validate our results with the application of performance-based facial capture using a recently-proposed anatomical local face deformation model

A Computational Framework for A First-Order System of Conservation Laws in Thermoelasticity

It is evidently not trivial to analytically solve practical engineering problems due to their inherent (geometrical and/or material) nonlinearities. Moreover, experimental testing can be extremely costly, time-consuming and even dangerous, in some cases. 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 a cost-effective manner.An important feature of a numerical methodology is how to approximate a physical domain into a computational domain and that, typically, can be carried out via mesh-based and particle-based approximations, either of which manifest with a different range of capabilities. Due to the geometrical complexity of many industrial applications (e.g. biomechanics, shape casting, metal forming, additive manufactur-ing, crash simulations), a growing attraction has been received by tetrahedral mesh generation, thanks to Delaunay and advancing front techniques [1, 2]. Alternatively, particle-based methods can be used as they offer the possibility of tackling specific applications in which mesh-based techniques may not be efficient (e.g. hyper velocity impact, astrophysics, failure simulations, blast).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 such as reduced order of convergence for strains and stresses in comparison with displacements and the possible onset of numerical instabilities (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 isothermal elastodynamics was presented in [3–6], in terms of the linear momentum p per unit undeformed volume and the minors of the deformation, namely, the deformation gradient F , its co-factor H and its Jacobian J. Taking inspiration of these works [4, 7] and in order to account for irreversible processes, the balance of total energy (also known as the first law of thermodynamics) is incorporated to the set of physical laws used to describe the deformation process. This, in general, can be expressed in terms of the entropy density η or total energy density E by which the Total Lagrangian entropy-based and total energy-based formulations {p, F , H, J, η or E} are established, respectively. Interestingly, taking advantage of the conservation formulation framework, it is possible to bridge the gap between solid dynamics and Computational Fluid Dynamics (CFD) by exploiting available CFD techniques in the context of solid dynamics.From a computational standpoint, two distinct and extremely competitive spatial discretisations are employed, namely, mesh-based Vertex-Centred Finite Volume Method (VCFVM) and meshless 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, respectively. Crucially, the discontinuous solution for the conservation variables across (dual) control volume interfaces or between any pair of particles is approximated via an acoustic Riemann solver. In addition, a tailor-made artificial compressibility algorithm and an angular momentum preservation scheme are also incorporated in order to assess same limiting scenarios.The semi-discrete system of equations is then temporally discretised using a one-step two-stage Total Variation Diminishing (TVD) Runge-Kutta time integrator, providing second order accuracy in time. The geometry is also monolithically updated to be only used for post-processing purposes.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 thermo-elasticity in comparison with classical finite element displacement-based approaches. The proposed computational framework provides a good balance between accuracy and speed of computation