The Penalty-Free Nitsche Method and Nonconforming Finite Elements for the Signorini Problem

We design and analyse a Nitsche method for contact problems. Compared to the seminal work of Chouly and Hild [SIAM J. Numer. Anal., 51 (2013), pp. 1295--1307], our method is constructed by expressing the contact conditions in a nonlinear function for the displacement variable instead of the lateral forces. The contact condition is then imposed using the nonsymmetric variant of Nitsche's method that does not require a penalty term for stability. Nonconforming piecewise affine elements are considered for the bulk discretization. We prove optimal error estimates in the energy norm

A CutFEM method for Stefan-Signorini problems with application in pulsed laser ablation

In this article, we develop a cut finite element method for one-phase Stefan problems, with applications in laser manufacturing. The geometry of the workpiece is represented implicitly via a level set function. Material above the melting/vaporisation temperature is represented by a fictitious gas phase. The moving interface between the workpiece and the fictitious gas phase may cut arbitrarily through the elements of the finite element mesh, which remains fixed throughout the simulation, thereby circumventing the need for cumbersome re-meshing operations. The primal/dual formulation of the linear one-phase Stefan problem is recast into a primal non-linear formulation using a Nitsche-type approach, which avoids the difficulty of constructing inf-sup stable primal/dual pairs. Through the careful derivation of stabilisation terms, we show that the proposed Stefan-Signorini-Nitsche CutFEM method remains stable independently of the cut location. In addition, we obtain optimal convergence with respect to space and time refinement. Several 2D and 3D examples are proposed, highlighting the robustness and flexibility of the algorithm, together with its relevance to the field of micro-manufacturing

A stable cut finite element method for multiple unilateral contact

International audienceThis paper presents a novel CutFEM-LaTIn algorithm to solve multiple unilateral contact problems over geometries that do not conform with the finite element mesh. We show that our method is (i) stable, independently of the interface locations (ii) optimally convergent with mesh refinement and (iii) efficient from an algorithmic point of view

A reduced basis method for frictional contact problems formulated with Nitsche's method

We develop an efficient reduced basis method for the frictional contact problem formulated using Nitsche's method. We focus on the regime of small deformations and on Tresca friction. The key idea ensuring the computational efficiency of the method is to treat the nonlinearity resulting from the contact and friction conditions by means of the Empirical Interpolation Method. The proposed algorithm is applied to the Hertz contact problem between two half-disks with parameter-dependent radius. We also highlight the benefits of the present approach with respect to the mixed (primal-dual) formulation

Fundamentally New Coupled Approach to Contact Mechanics via the Dirichlet-Neumann Schwarz Alternating Method

Contact phenomena are essential in understanding the behavior of mechanical systems. Existing computational approaches for simulating mechanical contact often encounter numerical issues, such as inaccurate physical predictions, energy conservation errors, and unwanted oscillations. We introduce an alternative technique, rooted in the non-overlapping Schwarz alternating method, originally developed for domain decomposition. In multi-body contact scenarios, this method treats each body as a separate, non-overlapping domain and prevents interpenetration using an alternating Dirichlet-Neumann iterative process. This approach has a strong theoretical foundation, eliminates the need for contact constraints, and offers flexibility, making it well-suited for multiscale and multi-physics applications. We conducted a numerical comparison between the Schwarz method and traditional methods like Lagrange multiplier and penalty methods, focusing on a benchmark impact problem. Our results indicate that the Schwarz alternating method surpasses traditional methods in several key areas: it provides more accurate predictions for various measurable quantities and demonstrates exceptional energy conservation capabilities. To address the issue of unwanted oscillations in contact velocities and forces, we explored various algorithms and stabilization techniques, ultimately opting for the naive-stabilized Newmark scheme for its simplicity and effectiveness. Furthermore, we validated the efficiency of the Schwarz method in a three-dimensional impact problem, highlighting its innate capacity to accommodate different mesh topologies, time integration schemes, and time steps for each interacting body

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Efficient computation and applications of the Calderón projector

The boundary element method (BEM) is a numerical method for the solution of partial differential equations through the discretisation of associated boundary integral equations.BEM formulations are commonly derived from properties of the Calderón projector, a blocked operator containing four commonly used boundary integral operators. In this thesis, we look in detail at the Calderón projector, derive and analyse a novel use of it to impose a range of boundary conditions, and look at how it can be efficiently computed. Throughout, we present computations made using the open-source software library Bempp, many features of which have been developed as part of this PhD. We derive a method for weakly imposing boundary conditions on BEM, inspired by Nitsche’s method for finite element methods. Formulations for Laplace problems with Dirichlet, Neumann, Robin, and mixed boudary conditions are derived and analysed. For Robin and mixed boundary conditions, the resulting formulations are simpler than standard BEM formulations, and convergence at a similar rate to standard methods is observed. As a more advanced application of this method, we derive a BEM formulation for Laplace’s equation with Signorini contact conditions. Using the weak imposition framework allows us to naturally impose this more complex boundary condition; the ability to do this is a significant advantage of this work. These formulations are derived and analysed, and numerical results are presented. Using properties of the Calderón projector, methods of operator preconditioning for BEM can be derived. These formulations involve the product of boundary operators. We present the details of a discrete operator algebra that allows the easy calculation of these products on the discrete level. This operator algebra allows for the easy implementation of various formulations of Helmholtz and Maxwell problems, including regularised combined field formulations that are immune to ill-conditioning near eigenvalues that are an issue for other formulations. We conclude this thesis by looking at weakly imposing Dirichlet and mixed Dirichlet–Neumann boundary condition on the Helmholtz equation. The theory for Laplace problems is extended to apply to Helmholtz problems, and an application to wave scattering from multiple scatterers is presented

Nonconforming boundary elements and finite elements for interface and contact problems with friction : hp-version for mortar, penalty and Nitsche's methods

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