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We present a practical algorithm for partially relaxing multiwell energy densities such as pertain to materials undergoing martensitic phase transitions. The algorithm is based on sequential lamination, but the evolution of the…

Materials Science · Physics 2015-06-24 S. Aubry , M. Fago , M. Ortiz

We consider the aeroelastic simulation of flexible mechanical structures submerged in subsonic fluid flows at low Mach numbers. The nonlinear kinematics of flexible bodies are described in the total Lagrangian formulation and discretized by…

Numerical Analysis · Mathematics 2024-03-25 Jenny Schubert , Marc C. Steinbach , Christian Hente , David Märtins , Daniel Schuster

We propose a deep neural network (DNN) as a fast surrogate model for local stress (and in principle strain) calculation in inhomogeneous non-linear material systems. We show that the DNN predicts the local stresses with about 3.8% mean…

Materials Science · Physics 2021-03-17 Jaber Rezaei Mianroodi , Nima H. Siboni , Dierk Raabe

The development of accurate constitutive models for materials that undergo path-dependent processes continues to be a complex challenge in computational solid mechanics. Challenges arise both in considering the appropriate model assumptions…

Machine Learning · Computer Science 2023-02-22 Jan N. Fuhg , Craig M. Hamel , Kyle Johnson , Reese Jones , Nikolaos Bouklas

Finite element methods usually construct basis functions and quadrature rules for multidimensional domains via tensor products of one-dimensional counterparts. While straightforward, this approach results in integration spaces larger than…

Numerical Analysis · Mathematics 2026-01-09 Tomas Teijeiro , Pouria Behnoudfar , Jamie M. Taylor , David Pardo , Victor M. Calo

Soft materials such as rubber and hydrogels are commonly used in industry for their excellent hyperelastic behaviour. There are various types of constitutive models for soft materials, and phenomenological models are very popular for finite…

Soft Condensed Matter · Physics 2021-06-29 Shun Meng , Haroon Imtiaz , Bin Liu

When discretizing symmetric stress tensors in variational problems arising in continuum mechanics, one has to choose how to enforce the symmetry of the stress tensor: (i) strongly by requiring the discrete tensors to be pointwise symmetric…

Numerical Analysis · Mathematics 2026-05-21 Pablo Brubeck , Charles Parker , Umberto Zerbinati

Topology optimization (TO) has experienced a dramatic development over the last decades aided by the arising of metamaterials and additive manufacturing (AM) techniques, and it is intended to achieve the current and future challenges. In…

For adaptive mixed finite element methods (AMFEM), we first introduce the data oscillation to analyze, without the restriction that the inverse of the coefficient matrix of the partial differential equations (PDEs) is a piecewise polynomial…

Numerical Analysis · Mathematics 2011-01-07 Shaohong Du , Xiaoping Xie

We introduce conformal mixed finite element methods for $2$D and $3$D incompressible nonlinear elasticity in terms of displacement, displacement gradient, the first Piola-Kirchhoff stress tensor, and pressure, where finite elements for the…

Numerical Analysis · Mathematics 2019-10-31 Arzhang Angoshtari

This work proposes a new efficient approach for calculating the bending stiffness of two-dimensional materials using simple atomistic tests on small periodic unit cells. The tests are designed such that bending deformations are dominating…

Materials Science · Physics 2022-12-23 Farzad Shirazian , Roger A. Sauer

We introduce a new class of mixed finite element methods for 2D and 3D compressible nonlinear elasticity. The independent unknowns of these conformal methods are displacement, displacement gradient, and the first Piola-Kirchhoff stress…

Numerical Analysis · Mathematics 2019-10-22 Arzhang Angoshtari , Ali Gerami Matin

The increasing availability of full-field displacement data from imaging techniques in experimental mechanics is determining a gradual shift in the paradigm of material model calibration and discovery, from using several simple-geometry…

Computational Engineering, Finance, and Science · Computer Science 2025-07-01 Saeid Ghouli , Moritz Flaschel , Siddhant Kumar , Laura De Lorenzis

This paper provides a new avenue for exploiting deep neural networks to improve physics-based simulation. Specifically, we integrate the classic Lagrangian mechanics with a deep autoencoder to accelerate elastic simulation of deformable…

Machine Learning · Computer Science 2021-02-23 Siyuan Shen , Yang Yin , Tianjia Shao , He Wang , Chenfanfu Jiang , Lei Lan , Kun Zhou

This paper is devoted to the exploration of rectangular finite elements' ability to model the stress-strain state of isotropic and orthotropic materials with a negative Poisson's ratio, known as auxetic materials. By employing linear…

Computational Physics · Physics 2025-05-13 A. V. Mazaev

The Tangential-Displacement Normal-Normal-Stress (TDNNS) method is a finite element method for mixed elasticity. As the name suggests, the tangential component of the displacement vector as well as the normal-normal component of the stress…

Numerical Analysis · Mathematics 2018-07-31 Astrid S. Pechstein , Joachim Schöberl

In this thesis we develop a stabilised finite element method for solving the equations of poroelasticity to enable solving complex models of biological tissues such as the human lungs. For the proposed numerical scheme, we use the lowest…

Numerical Analysis · Mathematics 2016-09-23 Lorenz Berger

Stress analysis of heterogeneous media, like composite materials, using Finite Element Analysis (FEA) has become commonplace in design and analysis. However, determining stress distributions in heterogeneous media using FEA can be…

Applied Physics · Physics 2021-04-22 Haotian Feng , Pavana Prabhakar

This work develops an elasto-plastic cell-based smoothed finite element method (CSFEM) for geotechnical analysis. The formulation incorporates a smoothed strain field into the standard elasto-plastic framework based on the Mohr-Coulomb…

Numerical Analysis · Mathematics 2025-11-26 Yang Yang , Mingjiao Yan , Zongliang Zhang , Miao Zhang , Feidong Zheng , Dong Pan , Xiaozi Lin

Developing engineering systems that rely on flow-induced reconfiguration, the phenomenon where a structure deforms under flow to reduce its drag, requires design tools that can predict the behavior of these flexible structures. Current…

Materials Science · Physics 2024-06-11 Danick Lamoureux , Sophie Ramananarivo , David Melancon , Frédérick P. Gosselin