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Tom Gustafsson

Publications and source records attributed to Tom Gustafsson.

At least 19 recordsLinked to original sources
arXiv2026

Nitsche methods for constrained problems in mechanics

We present guidelines for deriving new Nitsche Finite Element Methods to enforce equality and inequality constraints that act on the value of the unknown mechanical quantity. We first formulate the problem as a stabilized finite element method for the saddle point formulation where a Lagrange multiplier enforces the underlying constraint. The Nitsche method is then presented in a general minimization form, suitable for adding constraints to nonlinear finite element methods and allowing straightforward computational implementation with automatic differentation. This extends the method beyond classical boundary condition enforcement. To validate these ideas, we present Nitsche formulations for a range of problems in solid mechanics and give numerical evidence of the convergence rates of the Nitsche method.

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arXiv2025

Finite Element Methods for Elastic Contact: Penalty and Nitsche

We consider two methods for treating elastic contact problems with the finite element method; the penalty method and Nitsche's method. For the penalty method we discuss how the penalty parameter should be chosen. Both the theoretical analysis and numerical examples show that an optimal convergence rate cannot be achieved. The method is contrasted to that of Nitsche which is optimally convergent. We also give the derivation of Nitsche's method by a very simple consistency correction of the penalty method.

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arXiv2025

Hybrid Nitsche method for distributed computing

We extend a distributed finite element method built upon model order reduction to arbitrary polynomial degree using a hybrid Nitsche scheme. The new method considerably simplifies the transformation of the finite element system to the reduced basis for large problems. We prove that the error of the reduced Nitsche solution converges optimally with respect to the approximation order of the finite element spaces and linearly with respect to the dimension reduction parameter. Numerical tests with nontrivial tetrahedral meshes using second-degree polynomial bases support the theoretical results.

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arXiv2024

Adaptive finite elements for obstacle problems

We summarise three applications of the obstacle problem to membrane contact, elastoplastic torsion and cavitation modelling, and show how the resulting models can be solved using mixed finite elements. It is challenging to construct fixed computational meshes for any inequality-constrained problem because the coincidence set has an unknown shape. Consequently, we demonstrate how $h$-adaptivity can be used to resolve the unknown coincidence set. We demonstrate some practical challenges that must be overcome in the application of the adaptive method.

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arXiv2024

Distributed finite element solution using model order reduction

We extend a localized model order reduction method for the distributed finite element solution of elliptic boundary value problems in the cloud. We give a computationally efficient technique to compute the required inner product matrices and optimal reduced bases. A memory-efficient methodology is proposed to project the global finite element linear system onto the reduced basis. Our numerical results demonstrate the technique using non-trivial tetrahedral meshes and subdomain interfaces with up to 85 million degrees-of-freedom on a laptop computer by distributing the bulk of the model order reduction to the cloud.

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arXiv2023

Stabilised finite element method for Stokes problem with nonlinear slip condition

This work introduces a stabilised finite element formulation for the Stokes flow problem with a nonlinear slip boundary condition of friction type. The boundary condition is enforced with the help of an additional Lagrange multiplier and the stabilised formulation is based on simultaneously stabilising both the pressure and the Lagrange multiplier. We establish the stability and the a priori error analyses, and perform a numerical convergence study in order to verify the theory.

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arXiv2022

Mortaring for linear elasticity using mixed and stabilized finite elements

The purpose of this work is to study mortar methods for linear elasticity using standard low order finite element spaces. Based on residual stabilization, we introduce a stabilized mortar method for linear elasticity and compare it to the unstabilized mixed mortar method. For simplicity, both methods use a Lagrange multiplier defined on a trace mesh inherited from one side of the interface only. We derive a quasi-optimality estimate for the stabilized method and present the stability criteria of the mixed $P_1-P_1$ approximation. Our numerical results demonstrate the stability and the convergence of the methods for tie contact problems. Moreover, the results show that the mixed method can be successfully extended to three dimensional problems.

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arXiv2022

Mixed finite elements for Bingham flow in a pipe

We consider mixed finite element approximations of viscous, plastic Bingham flow in a cylindrical pipe. A novel a priori and a posteriori error analysis is introduced which is based on a discrete mesh dependent norm for the normalized Lagrange multiplier. This allows proving stability for various conforming finite elements. Numerical examples are presented to support the theory and to demonstrate adaptive mesh refinement.

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arXiv2021

Stabilized finite elements for Tresca friction problem

We formulate and analyze a Nitsche-type algorithm for frictional contact problems. The method is derived from, and analyzed as, a stabilized finite element method and shown to be quasi-optimal, as well as suitable as an adaptive scheme through an a posteriori error analysis. The a posteriori error indicators are validated in a numerical experiment.

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arXiv2021

A simple technique for unstructured mesh generation via adaptive finite elements

This work describes a concise algorithm for the generation of triangular meshes with the help of standard adaptive finite element methods. We demonstrate that a generic adaptive finite element solver can be repurposed into a triangular mesh generator if a robust mesh smoothing algorithm is applied between the mesh refinement steps. We present an implementation of the mesh generator and demonstrate the resulting meshes via examples.

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arXiv2020

Nitsche's method for Kirchhoff plates

We introduce a Nitsche's method for the numerical approximation of the Kirchhoff-Love plate equation under general Robin-type boundary conditions. We analyze the method by presenting a priori and a posteriori error estimates in mesh-dependent norms. Several numerical examples are given to validate the approach and demonstrate its properties.

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arXiv2020

On Nitsche's method for elastic contact problems

We show quasi-optimality and a posteriori error estimates for the frictionless contact problem between two elastic bodies with a zero-gap function. The analysis is based on interpreting Nitsche's method as a stabilised finite element method for which the error estimates can be obtained with minimal regularity assumptions and without the saturation assumption. We present three different Nitsche's mortaring techniques for the contact boundary each corresponding to a different stabilising term. Our numerical experiments show the robustness of Nitsche's method and corroborates the efficiency of the a posteriori error estimators.

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arXiv2019

Numerical modelling of coupled linear dynamical systems

Numerical modelling of several coupled passive linear dynamical systems (LDS) is considered. Since such component systems may arise from partial differential equations, transfer function descriptions, lumped systems, measurement data, etc., the first step is to discretise them into finite-dimensional LDSs using, e.g., the finite element method, autoregressive techniques, and interpolation. The finite-dimensional component systems may satisfy various types of energy (in)equalities due to passivity that require translation into a common form such as the scattering passive representation. Only then can the component systems be coupled in a desired feedback configuration by computing pairwise Redheffer star products of LDSs. Unfortunately, a straightforward approach may fail due to ill-posedness of feedback loops between component systems. Adversities are particularly likely if some component systems have no energy dissipation at all, and this may happen even if the fully coupled system could be described by a finite-dimensional LDS. An approach is proposed for obtaining the coupled system that is based on passivity preserving regularisation. Two practical examples are given to illuminate the challenges and the proposed methods to overcome them: the Butterworth low-pass filter and the termination of an acoustic waveguide to an irrational impedance.

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arXiv2018

Nitsche's method for unilateral contact problems

We derive optimal a priori and a posteriori error estimates for Nitsche's method applied to unilateral contact problems. Our analysis is based on the interpretation of Nitsche's method as a stabilised finite element method for the mixed Lagrange multiplier formulation of the contact problem wherein the Lagrange multiplier has been eliminated elementwise. To simplify the presentation, we focus on the scalar Signorini problem and outline only the proofs of the main results since most of the auxiliary results can be traced to our previous works on the numerical approximation of variational inequalities. We end the paper by presenting results of our numerical computations which corroborate the efficiency and reliability of the a posteriori estimators.

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arXiv2018

Error analysis of Nitsche's mortar method

Optimal a priori and a posteriori error estimates are derived for Nitsche's mortar finite elements. The analysis is based on the equivalence of the Nitsche's method and the stabilised mixed method. The Nitsche's method is defined so that it is robust with respect to large jumps in the material and mesh parameters over the interface. Numerical results demonstrate the robustness of the a posteriori estimators.

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arXiv2017

An adaptive finite element method for the inequality-constrained Reynolds equation

We present a stabilized finite element method for the numerical solution of cavitation in lubrication, modeled as an inequality-constrained Reynolds equation. The cavitation model is written as a variable coefficient saddle-point problem and approximated by a residual-based stabilized method. Based on our recent results on the classical obstacle problem, we present optimal a priori estimates and derive novel a posteriori error estimators. The method is implemented as a Nitsche-type finite element technique and shown in numerical computations to be superior to the usually applied penalty methods.

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arXiv2017

A posteriori estimates for conforming Kirchhoff plate elements

We derive a residual a posteriori estimator for the Kirchhoff plate bending problem. We consider the problem with a combination of clamped, simply supported and free boundary conditions subject to both distributed and concentrated (point and line) loads. Extensive numerical computations are presented to verify the functionality of the estimators.

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