Mortar Finite Element Research Articles

A Neumann–Neumann algorithm for a mortar finite element discretization of fourth‐order elliptic problems in 2D

AbstractHere we present and analyze a Neumann–Neumann algorithm for the mortar finite element discretization of elliptic fourth‐order problems with discontinuous coefficients. The fully parallel algorithm is analyzed using the abstract Schwarz framework, proving a convergence which is independent of the parameters of the problem, and depends only logarithmically on the ratio between the subdomain size and the mesh size.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Computational modeling of surface phenomena in soft-wet materials

We consider the problem of extracting tribological information from experimental observations of contact with soft-wet materials. Particular attention is placed on simulating the response of two rotating cylinders of soft specimens placed in frictional contact, with a variable coefficient of friction dependent on the relative sliding velocity. The bulk behavior is modeled by means of a finite deformation viscoelasticity formulation, with constitutive parameters taken to be representative of hydrogels. We focus on the modeling of the surface behavior and employ a mortar-finite element contact formulation. Through a series of numerical studies, we demonstrate the strong sensitivity of the results to the choice of interfacial constitutive parameters. The difficulties of extracting such parameters using only experimental data and approximate analytical expressions are also examined.

Prediction of flow induced noise in rotating devices using nonmatching grids

With increasing number of electrical devices, e.g. air conditioning systems, used in homes and offices, noise pollution is becoming a more and more relevant topic. A large amount of this noise is generated by turbulent flows and laminar flows at leading and trailing edges, where mainly tonal noise is generated. The objective of our contribution is to investigate shape optimizations of rotating devices in order to reduce their noise levels. For this purpose, we conduct a simulation of the turbulent flow in a ventilator. The acoustic source terms are obtained from the fluid dynamics solution by using Lighthill's acoustic analogy. The acoustic domain is decomposed into a rotating part and a fixed part. The coupling between these two parts is enforced at their interface by a mortar finite element method, which uses Lagrange multipliers in order to ''glue'' the geometrically independent parts together. The mortar method takes into account the movement of the rotating part through a moving nonmatching grid, that is recomputed at each time step.

Mortar finite element discretization of a model coupling Darcy and Stokes equations

As a first draft of a model for a river flowing on a homogeneous porous ground, we consider a system where the Darcy and Stokes equations are coupled via appropriate matching conditions on the interface. We propose a discretization of this problem which combines the mortar method with standard finite elements, in order to handle separately the flow inside and outside the porous medium. We prove a priori and a posteriori error estimates for the resulting discrete problem. Some numerical experiments confirm the interest of the discretization.

Mortar element methods for parabolic problems

AbstractIn this article a standard mortar finite element method and a mortar element method with Lagrange multiplier are used for spatial discretization of a class of parabolic initial‐boundary value problems. Optimal error estimates in L∞(L2) and L∞(H1)‐norms for semidiscrete methods for both the cases are established. The key feature that we have adopted here is to introduce a modified elliptic projection. In the standard mortar element method, a completely discrete scheme using backward Euler scheme is discussed and optimal error estimates are derived. The results of numerical experiments support the theoretical results obtained in this article. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008

Coupling Discontinuous Galerkin and Mixed Finite Element Discretizations using Mortar Finite Elements

Discontinuous Galerkin (DG) and mixed finite element (MFE) methods are two popular methods that possess local mass conservation. In this paper we investigate DG-DG and DG-MFE domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface. The subdomain grids need not match, and the mortar grid may be much coarser, giving a two-scale method. Convergence results in terms of the fine subdomain scale $h$ and the coarse mortar scale $H$ are established for both types of couplings. In addition, a nonoverlapping parallel domain decomposition algorithm is developed, which reduces the coupled system to an interface mortar problem. The properties of the interface operator are analyzed.

A mortar method for thermomechanically coupled problems

AbstractIn the foundry industry a reliable prediction of quality defects of the products like residual stresses or size alterations requires detailed constitutive material equations along with a precise modelling of the cast/mould interaction. In this contribution we focus on the description of a contact‐dependent heat transfer between the metal and the mould. Within this approach the contact formulation is based on the mortar finite element scheme. This strategy has turned out to be more stable for contact problems than other schemes like node‐to‐segment discretizations, see e.g.[1]. Due to the significant difference in the stiffness of the mould and the cast in the liquid or semi‐solid state a formulation is proposed that works without additional parameters like a penalty parameter or Lagrange multipliers. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Mortar finite volume element method with Crouzeix–Raviart element for parabolic problems

In this paper, we consider a semi-discrete mortar finite volume element method for two-dimensional parabolic problems. This method is based on the mortar Crouzeix–Raviart non-conforming finite element space. It is proved that the mortar finite volume element approximations derived are convergent with the optimal order in the H 1 - and L 2 -norms. Numerical experiments are presented to illustrate the theoretical results.

A MULTISCALE MORTAR MIXED FINITE ELEMENT METHOD FOR SLIGHTLY COMPRESSIBLE FLOWS IN POROUS MEDIA

We consider multiscale mortar mixed finite element discretizations for slightly compressible Darcy flows in porous media. This paper is an extension of the formulation introduced by Arbogast et al. for the incompressible problem [2]. In this method, flux continuity is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. Optimal fine scale convergence is obtained by an appropriate choice of mortar grid and polynomial degree of approximation. Parallel numerical simulations on some multiscale benchmark problems are given to show the efficiency and effectiveness of the method.

A new minimization protocol for solving nonlinear Poisson–Boltzmann mortar finite element equation

The nonlinear Poisson–Boltzmann equation (PBE) is a widely-used implicit solvent model in biomolecular simulations. This paper formulates a new PBE nonlinear algebraic system from a mortar finite element approximation, and proposes a new minimization protocol to solve it efficiently. In particular, the PBE mortar nonlinear algebraic system is proved to have a unique solution, and is equivalent to a unconstrained minimization problem. It is then solved as the unconstrained minimization problem by the subspace trust region Newton method. Numerical results show that the new minimization protocol is more efficient than the traditional merit least squares approach in solving the nonlinear system. At least 80 percent of the total CPU time was saved for a PBE model problem.

An a Posteriori Error Estimator for Two-Body Contact Problems on Non-Matching Meshes

A posteriori error estimates for two-body contact problems are established. The discretization is based on mortar finite elements with dual Lagrange multipliers. To define locally the error estimator, Arnold---Winther elements for the stress and equilibrated fluxes for the surface traction are used. Using the Lagrange multiplier on the contact zone as Neumann boundary conditions, equilibrated fluxes can be locally computed. In terms of these fluxes, we define on each element a symmetric and globally H(div)-conforming approximation for the stress. Upper and lower bounds for the discretization error in the energy norm are provided. In contrast to many other approaches, the constant in the upper bound is, up to higher order terms, equal to one. Numerical examples illustrate the reliability and efficiency of the estimator.

A priori error analysis of the mortar finite element method for variational inequalities

AbstractThe mortar finite element method is a special domain decomposition method, which can handle the situation where meshes on different subdomains need not align across the interface. In this article, we will apply the mortar element method to general variational inequalities of free boundary type, such as free seepage flow, which may show different behaviors in different regions. We prove that if the solution of the original variational inequality belongs to H2(D), then the mortar element solution can achieve the same order error estimate as the conforming P1 finite element solution. Application of the mortar element method to a free surface seepage problem and an obstacle problem verifies not only its convergence property but also its great computational efficiency. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Efficient Linear Solvers for Mortar Finite-Element Method

Efficient linear solvers for mortar finite element method are studied. An analysis of a brushless DC motor shows that proposed preconditioners improve the convergence to the solutions of linear systems with and without Lagrange multipliers

A Multiscale Mortar Mixed Finite Element Method

We develop multiscale mortar mixed finite element discretizations for second order elliptic equations. The continuity of flux is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. The polynomial degree of the mortar and subdomain approximation spaces may differ; in fact, the mortar space achieves approximation comparable to the fine scale on its coarse grid by using higher order polynomials. Our formulation is related to, but more flexible than, existing multiscale finite element and variational multiscale methods. We derive a priori error estimates and show, with appropriate choice of the mortar space, optimal order convergence and some superconvergence on the fine scale for both the solution and its flux. We also derive efficient and reliable a posteriori error estimators, which are used in an adaptive mesh refinement algorithm to obtain appropriate subdomain and mortar grids. Numerical experiments are presented in confirmation of the theory.

Mortar finite element discretization for the flow in a nonhomogeneous porous medium

Darcy’s equations model the flow of a viscous incompressible fluid in a rigid porous medium. One of the parameters of the system depends on the permeability of the medium and, when this medium is not homogeneous, the variations of the parameter could be very high. To handle this phenomenon, we propose a discretization of the model that relies on the mortar finite element method. Indeed, the idea is to construct a decomposition of the domain such that the permeability is constant on each element of the partition and to use independent meshes on the different subdomains. We perform the a priori and a posteriori analysis of this discretization and present some numerical experiments which are in good coherency with the results of the analysis.

Stable Lagrange multipliers for quadrilateral meshes of curved interfaces in 3D

For the discretization of coupled problems by means of mortar finite elements, the choice of the discrete Lagrange multiplier space is of crucial importance for the performance of the resulting numerical scheme. Dual Lagrange multipliers offer several advantages since they can be locally eliminated. In this paper, the lowest order dual Lagrange multipliers are extended to arbitrary quadrilateral or triangular surface grids. In a first step, two alternatives are provided for the case of planar interfaces. For the approximation of vector fields, two more modifications are given for use on curvilinear interfaces. A priori results are obtained and several numerical examples are shown.

Biorthogonal bases with local support and approximation properties

We construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension. In contrast to earlier approaches, these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p - 1. Working with Gaus-Lobatto nodes, we find an interesting connection between biorthogonality and quadrature formulas. One important application of these newly constructed biorthogonal basis functions are two-dimensional mortar finite elements. The weak continuity condition of the constrained mortar space is realized in terms of our new dual bases. As a result, local static condensation can be applied which is very attractive from the numerical point of view. Numerical results are presented for cubic mortar finite elements.

On the accuracy of the mortar finite volume element method with Lagrange multipliers

In this paper, the mortar finite volume element (MFVE) method with Lagrange multipliers is considered for second order elliptic boundary value problems. We prove L 2 error estimates which are affected by regularities in both the exact solution and the source term. Meanwhile, the uniform convergence of the solution is obtained for the MFVE method under minimal regularity assumption.

Elasto–acoustic and acoustic–acoustic coupling on non‐matching grids

AbstractFlexible discretization techniques for the approximative solution of coupled wave propagation problems are investigated, focussing on aero–acoustic and elasto–acoustic coupling. In particular, the advantages of using non‐matching grids are presented, when one subregion has to be resolved by a substantially finer grid than the other subregion. For the elasto–acoustic coupling, the problem formulation remains essentially the same as for the matching situation, while for the aero–acoustic coupling, the formulation is enhanced with Lagrange multipliers within the framework of mortar finite element methods. Several numerical examples are presented to demonstrate the flexibility and applicability of the approach. Copyright © 2006 John Wiley & Sons, Ltd.

Two‐Level Schwarz Algorithms with Overlapping Subregions for Mortar Finite Elements

Preconditioned conjugate gradient methods based on two-level overlapping Schwarz methods often perform quite well. Such a preconditioner combines a coarse space solver with local components which are defined in terms of subregions that form an overlapping covering of the region on which the elliptic problem is defined. Precise bounds on the rate of convergence of such iterative methods have previously been obtained in the case of conforming lower order and spectral finite elements as well as in a number of other cases. In this paper, this domain decomposition algorithm and analysis are extended to mortar finite elements. It is established that the condition number of the relevant iteration operator is independent of the number of subregions and varies with the relative overlap between neighboring subregions linearly as in the conforming cases previously considered.