arXiv subjects

Erik Faust

Publications and source records attributed to Erik Faust.

8 recordsLinked to original sources

arXiv2026

Component-wise hyperreduction for nonlinear solid mechanics problems

Hyperreduced nonlinear solid-mechanics components can be generated offline and reused as transferable building blocks across different assemblies, boundary conditions, meshes, material parameters, and constitutive models. We use proper orthogonal decomposition (POD) and energy conserving sampling and weighting (ECSW) on the component level and connect the substructures by mortar mesh tying. The POD modes and the ECSW weights and elements are computed offline from simulations of single components and the finite rigid body motions are treated by 12 additional rigid body modes per substructure. The numerical examples demonstrate errors below 1 \% for large quasi-static assemblies while evaluating less than 10 \% of the elements. The same component bases and ECSW elements are successfully reused for finite-strain viscoelastic dynamics, although they were trained only on elastic Neo-Hookean component simulations. These results indicate that component-wise hyperreduction can provide reusable reduced building blocks for modular nonlinear solid-mechanics simulations.

cs.CE↗

arXiv2026

Efficient strain-space hyperreduction in large-deformation solid mechanics

Strain-space model order reduction (MOR) techniques have recently been shown to achieve exceptional performance in terms of the tradeoff between runtime and accuracy achieved in computational homogenisation problems. In this article, we generalise such techniques to problems in large-deformation solid mechanics beyond the context of computational homogenisation. Arbitrary-valued, parameterised Dirichlet boundary conditions are satisfied by construction using a lifting with boundary-consistent fields computed offline. This allows us to pose a version of the Empirical Cubature Method (ECM) [24,25] in strain space and generalise the Empirically Corrected Cluster Cubature (E3C) [46,48,49] as well as Empirical Material Sampling and Linearisation (EMSL) [17] beyond computational homogenisation problems. The strain-space versions of EMSL, ECM, and E3C are compared against each other and a standard displacement-space formulation of Energy Conserving Weighting and Sampling (ECSW) [15]. On two hyperelastic example problems with parameterised material behaviour and deformation, the strain-space methods outperform the displacement-space alternative in the tradeoff between runtime and accuracy. E3C and EMSL in particular facilitate 10,000 and 100,000-fold speedups, respectively, while retaining high levels of accuracy. EMSL is shown to be the method of choice when online and offline runtime budgets are very limited, while E3C yields exceptional levels of accuracy when slightly more runtime is acceptable.

cs.CE↗

arXiv2026

Empirical Material Sampling and Linearisation -- A Simple and Efficient Strain-Space Model Order Reduction Approach for Computational Homogenisation in Large-Deformation Hyperelasticity

In this article, we propose a simple and efficient hyperreduced strain-space model order reduction (MOR) approach for hyperelastic representative volume elements (RVEs), called Empirical Material Sampling and Linearisation (EMSL). The approach is conceptually motivated by the Empirically Corrected Cluster Cubature (E3C) of Wulfinghoff and Hauck [36], but also draws on ideas from previous work on incremental variational structure-preserving strain-space model order reduction techniques to achieve rapid evaluations in the online phase. As in E3C, we group the material domain into regions of similar behaviour, and query the material routine at one reference strain value per region. However, we sample these strains only once per load increment, at empirically estimated expected strain values. We use the reference material tangent and strain modes obtained via the Proper Orthogonal Decomposition (POD) to compute a linearised estimate of the stress response in the remainder of the material cluster. In contrast to E3C, which approximately integrates the exact material law, EMSL could therefore be said to exactly integrate an approximation of the material behaviour. The resulting reduced problem is affine in each load step, allowing for integration over the entire computational domain via operations which can readily be preprocessed in the offline phase. Since a linear equation system is obtained in each load increment, no Newton iterations are required in the online phase. For benchmark comparisons, we pose a variant of two popular reduced cubature schemes in strain space and recall the E3C algorithm proposed by Wulfinghoff et al. On an example hyperelastic RVE problem with a porous geometry, we show that EMSL Pareto-dominates competing strain-space approaches in terms of the tradeoff between accuracy and runtime.

cs.CE↗

arXiv2025

A hyperreduced manifold learning approach to nonlinear model order reduction for the homogenisation of hyperelastic RVEs

In a recent work, we proposed a graph-based manifold learning scheme for the nonlinear Galerkin-reduction of quasi-static solid mechanical problems [1]. The resulting nonlinear approximation spaces can closely and flexibly represent nonlinear solution manifolds. The present work discusses how this nonlinear model order reduction (MOR) approach can be employed to reduce online computational costs by multiple orders of magnitude while retaining high levels of accuracy. We integrate two popular hyperreduction methods into the nonlinear MOR framework and discuss how we achieve an algorithmic complexity which is independent from the original system size. Furthermore, improvements are made to the local online linearisation scheme for the sake of performance and robustness. On an example RVE problem, the MOR scheme accelerates computations by more than two orders of magnitude with little training data and negligible loss of accuracy. Additionally, the algorithm Pareto-dominates alternative approaches in the trade-off between accuracy and runtime on the considered example.

cs.CE↗

arXiv2025

Extending the Lattice Boltzmann Method to Non-linear Solid Mechanics

This work outlines a Lattice Boltzmann Method (LBM) for geometrically and constitutively nonlinear solid mechanics to simulate large deformations under dynamic loading conditions. The method utilizes the moment chain approach, where the non-linear constitutive law is incorporated via a forcing term. Stress and deformation measures are expressed in the reference configuration. Finite difference schemes are employed for gradient and divergence computations, and Neumann- and Dirichlet-type boundary conditions are introduced. Numerical studies are performed to assess the proposed method and illustrate its capabilities. Benchmark tests for weakly dynamic uniaxial tension and simple shear across a range of Poisson's ratios demonstrate the feasibility of the scheme and serve as validation of the implementation. Furthermore, a dynamic test case involving the propagation of bending waves in a cantilever beam highlights the potential of the method to model complex dynamic phenomena.

cs.CE↗

arXiv2024

A manifold learning approach to nonlinear model order reduction of quasi-static problems in solid mechanics

The proper orthogonal decomposition (POD) -- a popular projection-based model order reduction (MOR) method -- may require significant model dimensionalities to successfully capture a nonlinear solution manifold resulting from a parameterised quasi-static solid-mechanical problem. The local basis method by Amsallem et al. [1] addresses this deficiency by introducing a locally, rather than globally, linear approximation of the solution manifold. However, this generally successful approach comes with some limitations, especially in the data-poor setting. In this proof-of-concept investigation, we instead propose a graph-based manifold learning approach to nonlinear projection-based MOR which uses a global, continuously nonlinear approximation of the solution manifold. Approximations of local tangents to the solution manifold, which are necessary for a Galerkin scheme, are computed in the online phase. As an example application for the resulting nonlinear MOR algorithms, we consider simple representative volume element computations. On this example, the manifold learning approach Pareto-dominates the POD and local basis method in terms of the error and runtime achieved using a range of model dimensionalities.

cs.CE↗

arXiv2022

A Lattice Boltzmann Method for Elastic Solids Under Plane Strain Deformation

The Lattice Boltzmann Method (LBM), e.g. in [ 1] and [2 ], can be interpreted as an alternative method for the numerical solution of partial differential equations. Consequently, although the LBM is usually applied to solve fluid flows, the above interpretation of the LBM as a general numerical tool, allows the LBM to be extended to solid mechanics as well. In this spirit, the LBM has been studied in recent years. First publications [3], [4] presented an LBM scheme for the numerical solution of the dynamic behavior of a linear elastic solid under simplified deformation assumptions. For so-called anti-plane shear deformation, the only non-zero displacement component is governed by a two-dimensional wave equation. In this work, an existing LBM for the two-dimensional wave equation is extended to more general plane strain problems. The proposed algorithm reduces the plane strain problem to the solution of two separate wave equations for the volume dilatation and the non-zero component of the rotation vector, respectively. A particular focus is on the implementation of types of boundary conditions that are commonly encountered in engineering practice for solids: Dirichlet and Neumann boundary conditions. Last, several numerical experiments are conducted that highlight the performance of the new LBM in comparison to the Finite Element Method.

math.NA↗

arXiv2022

Dirichlet and Neumann boundary conditions in a Lattice Boltzmann Method for Elastodynamics

Recently, Murthy et al. [2017] and Escande et al. [2020] adopted the Lattice Boltzmann Method (LBM) to model the linear elastodynamic behaviour of isotropic solids. The LBM is attractive as an elastodynamic solver because it can be parallelised readily and lends itself to finely discretised dynamic continuum simulations, allowing transient phenomena such as wave propagation to be modelled efficiently. This work proposes simple local boundary rules which approximate the behaviour of Dirichlet and Neumann boundary conditions with an LBM for elastic solids. Both lattice-conforming and non-lattice-conforming, curved boundary geometries are considered. For validation, we compare results produced by the LBM for the sudden loading of a stationary crack with an analytical solution. Furthermore, we investigate the performance of the LBM for the transient tension loading of a plate with a circular hole, using Finite Element (FEM) simulations as a reference.

cs.CE↗