FE/BE coupling for an acoustic fluid-structure interaction problem. Residual a posteriori error estimates

This is the author's accepted manuscript. The final published article is available from the link below. Copyright © 2011 John Wiley & Sons, Ltd.In this paper, we developed an a posteriori error analysis of a coupling of finite elements and boundary elements for a fluid–structure interaction problem in two and three dimensions. This problem is governed by the acoustic and the elastodynamic equations in time-harmonic vibration. Our methods combined integral equations for the exterior fluid and FEMs for the elastic structure. It is well-known that because of the reduction of the boundary value problem to boundary integral equations, the solution is not unique in general. However, because of superposition of various potentials, we consider a boundary integral equation that is uniquely solvable and avoids the irregular frequencies of the negative Laplacian operator of the interior domain. In this paper, two stable procedures were considered; one is based on the nonsymmetric formulation and the other is based on a symmetric formulation. For both formulations, we derived reliable residual a posteriori error estimates. From the estimators we computed local error indicators that allowed us to develop an adaptive mesh refinement strategy. For the two-dimensional case we performed an adaptive algorithm on triangles, and for the three-dimensional case we used hanging nodes on hexahedrons. Numerical experiments underline our theoretical results.DFG German Research Foundatio

Adaptive time domain boundary element methods and engineering applications

Time domain Galerkin boundary elements provide an efficient tool for the numerical solution of boundary value problems for the homogeneous wave equation. We review recent advances in their a posteriori error analysis and the resulting adaptive mesh refinement procedures, as well as basic algorithmic aspects of these methods. Numerical results for adaptive mesh refinements are discussed in 2 and 3 dimensions, as are benchmark problems in a half space related to the transient emission of traffic noise.Parts of this work were funded by BMWi under the project SPERoN 2020, part II, Leiser StraB enverkehr, grant number 19 U 10016 F. H. G. acknowledges support by ERC Advanced Grant HARG 268105

Adaptive FE-BE Coupling for Strongly Nonlinear Transmission Problems with Coulomb Friction

We analyze an adaptive finite element/boundary element procedure for scalar elastoplastic interface problems involving friction, where a nonlinear uniformly monotone operator such as the p-Laplacian is coupled to the linear Laplace equation on the exterior domain. The problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which is then solved using the Uzawa algorithm and adaptive mesh refinements based on a gradient recovery scheme. The Galerkin approximations are shown to converge to the unique solution of the variational problem in a suitable product of L^p- and L^2-Sobolev spaces.Comment: 27 pages, 3 figure

Dual-dual formulation for a contact problem with friction

A variational inequality formulation is derived for some frictional contact problems from linear elasticity. The formulation exhibits a two-fold saddle point structure and is of dual-dual type, involving the stress tensor as primary unknown as well as the friction force on the contact surface by means of a Lagrange multiplier. The approach starts with the minimization of the conjugate elastic potential. Applying Fenchel's duality theory to this dual minimization problem, the connection to the primal minimization problem and a dual saddle point problem is achieved. The saddle point problem possesses the displacement field and the rotation tensor as further unknowns. Introducing the friction force yields the dual-dual saddle point problem. The equivalence and unique solvability of both problems is shown with the help of the variational inequality formulations corresponding to the saddle point formulations, respectively.This work is supported by the German Research Foundation within the priority program 1180 Prediction and Manipulation of Interactions between Structure and Process

High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling

Copyright @ 2014 The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over inline image for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease

Quasi-optimal degree distribution for a quadratic programming problem arising from the p-version finite element method for a one-dimensional obstacle problem

We present a quadratic programming problem arising from the p-version for a finite element method with an obstacle condition prescribed in Gauss-Lobatto points. We show convergence of the approximate solution to the exact solution in the energy norm. We show an a-priori error estimate and derive an a-posteriori error estimate based on bubble functions which is used in an adaptive p-version. Numerical examples show the superiority of the p-version compared with the h-version. © 2013 Elsevier B.V. All rights reserved

A Boundary Element Procedure for 3D Electromagnetic Transmission Problems with Large Conductivity

Data Availability Statement: Not applicable.Copyright © 2022 by the authors. We consider the scattering of time-periodic electromagnetic fields by metallic obstacles, or the eddy current problem. In this interface problem, different sets of Maxwell equations must be solved both in the obstacle and outside it, while the tangential components of both electric and magnetic fields are continuous across the interface. We describe an asymptotic procedure, applied for large conductivity, which reflects the skin effect in metals. The key to our method is a special integral equation procedure for the exterior boundary value problems corresponding to perfect conductors. The asymptotic procedure leads to a great reduction in complexity for the numerical solution, since it involves solving only the exterior boundary value problems. Furthermore, we introduce a FEM/BEM coupling procedure for the transmission problem and consider the implementation of Galerkin’s elements for the perfect conductor problem, and present numerical experiments.ALECOL-DAAD Program and COLCIENCIAS through contract number CT 793-2013, and code 1215-569-33876

Residual-based a posteriori error estimate for hypersingular equation on surfaces

The hypersingular integral equation of the first kind equivalently describes screen and Neumann problems on an open surface piece. The paper establishes a computable upper error bound for its Galerkin approximation and so motivates adaptive mesh refining algorithms. Numerical experiments for triangular elements on a screen provide empirical evidence of the superiority of adapted over uniform mesh-refining. The numerical realisation requires the evaluation of the hypersingular integral operator at a source point; this and other details on the algorithm are included

Comparing induction at an early and late step in signal transduction mediating indirect defence in Brassica oleracea

The induction of plant defences involves a sequence of steps along a signal transduction pathway, varying in time course. In this study, the effects of induction of an early and a later step in plant defence signal transduction on plant volatile emission and parasitoid attraction are compared. Ion channel-forming peptides represent a class of inducers that induce an early step in signal transduction. Alamethicin (ALA) is an ion channel-forming peptide mixture from the fungus Trichoderma viride that can induce volatile emission and increase endogenous levels of jasmonic acid (JA) and salicylic acid in plants. ALA was used to induce an early step in the defence response in Brussels sprouts plants, Brassica oleracea var. gemmifera, and to study the effect on volatile emission and on the behavioural response of parasitoids to volatile emission. The parasitoid Cotesia glomerata was attracted to ALA-treated plants in a dose-dependent manner. JA, produced through the octadecanoid pathway, activates a later step in induced plant defence signal transduction, and JA also induces volatiles that are attractive to parasitoids. Treatment with ALA and JA resulted in distinct volatile blends, and both blends differed from the volatile blends emitted by control plants. Even though JA treatment of Brussels sprouts plants resulted in higher levels of volatile emission, ALA-treated plants were as attractive to C. glomerata as JA-treated plants. This demonstrates that on a molar basis, ALA is a 20 times more potent inducer of indirect plant defence than JA, although this hormone has more commonly been used as a chemical inducer of plant defence