Mortar method for matching grids in a mixed scheme as applied to the biharmonic equation

Nitsche’s mortar method for matching grids in the Hermann-Miyoshi mixed scheme for the biharmonic equation is considered. A two-parameter mortar problem is constructed and analyzed. Existence and uniqueness theorems are proved under certain constraints on the parameters. The norm of the difference between the solutions to the mortar and original problems is estimated. The convergence rates are the same as in the Hermann-Miyoshi scheme on matching grids.     

Nitsche type mortaring for elliptic problems with corner singularities

The paper deals with Nitsche type mortaring as a finite element method (FEM) for treating non‐matching meshes of triangles at the interface of some domain decomposition. The approach is applied to the Poisson equation with Dirichlet conditions for the case that the interface passes re‐entrant corners of the domain and local mesh refinement is applied. Some properties of the finite element scheme and error estimates in a discrete H¹‐like and in the L₂‐norm are proved.     

Stability of thin walled structures strongly coupled with contact

Stability problems strongly coupled with contact are examined using different contact algorithms. Applying the penalty formulation a high dependence of the buckling load on the penalty parameter is found. This observation becomes clear when considering a classical Euler case. Applying kinematical constraint conditions (close to a Lagrange scheme) for contact on the other hand more consistent results regarding the buckling load are obtained. Thus for a Nitsche scheme a better and more flexible solution is expected. The FE implementation of a Nitsche interface element is carried out. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak coupling of thin-walled multi-patch NURBS structures in the framework of isogeometric analysis

In the framework of isogeometric analysis, models are typically derived by the use of computer-aided geometric design (CAGD) tools which often results in a large number of non-conforming NURBS patches that are connected along arbitrary curved boundaries. A strong coupling on the basis of matching control meshes is rarely possible and limits the modeling process for practical applications. Weak coupling according to the principles introduced by Nitsche is the method of choice if a stable and variationally consistent method is favored which does not require the solution of additional equations to enforce the coupling constraints. We concentrate on the weak coupling of thin-walled shell structures modeled according to the theory of Kirchhoff-Love. The proposed concept is free of ad-hoc decisions for stabilization thus truly supporting a design-through-analysis idea. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal convergence rates for semidiscrete approximations of parabolic problems with nonsmooth boundary data

We consider semidiscrete approximations of parabolic boundary value problems based on an elliptic approximation by J. Nitsche, in which the approximating subspaces are not subject to any boundary conditions. Optimal L_p (L ₂) error estimates are derived for both smooth and nonsmooth boundary data. The approach is

based on semigroup theory combined with the theory of singular integrals.     

Aubin—Nitsche Estimates Are Equivalent to Compact Embeddings

In this short note we show that having an Aubin–Nitsche type estimate (a superconvergence estimate in a weaker norm for a convergent Galerkin method) is equivalent to having compact injection of the space into its completion with the weaker norm.     

Nitsche’s Method for a Transport Problem in Two-phase Incompressible Flows

We consider a parabolic interface problem which models the transport of a dissolved species in two-phase incompressible flow problems. Due to the so-called Henry interface condition the solution is discontinuous across the interface. We use an extended finite element space combined with a method due to Nitsche for the spatial discretization of this problem and derive optimal discretization error bounds for this method. For the time discretization a standard θ-scheme is applied. Results of numerical experiments are given that illustrate the convergence properties of this discretization.     

Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data

Summary Error estimates for the semidiserete Galerkin method for abstract semilinear evolution equations with non-smooth initial data are given. In concrete cases almost optimal order of convergence for linear finite elements results.To Professor Dr. J.A. Nitsche on the occasion of his sixtieth birthday     

Neuro-fuzzy based constraint programming

Constraint programming models appear in many sciences including mathematics, engineering and physics. These problems aim at optimizing a cost function joint with some constraints. Fuzzy constraint programming has been developed for treating uncertainty in the setting of optimization problems with vague constraints. In this paper, a new method is presented into creation fuzzy concept for set of constraints. Unlike to existing methods, instead of constraints with fuzzy inequalities or fuzzy coefficients or fuzzy numbers, vague nature of constraints set is modeled using learning scheme with adaptive neural-fuzzy inference system (ANFIS). In the proposed approach, constraints are not limited to differentiability, continuity, linearity; also the importance degree of each constraint can be easily applied. Unsatisfaction of each weighted constraint reduces membership of certainty for set of constraints. Monte-Carlo simulations are used for generating feature vector samples and outputs for construction of necessary data for ANFIS. The experimental results show the ability of the proposed approach for modeling constrains and solving parametric programming problems.     

Analysis of optimal preconditioners for CutFEM

In this article, we consider a class of unfitted finite element methods for scalar elliptic problems. These so-called CutFEM methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model problem we consider the application of such a method to the Poisson interface problem. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The unfitted finite element space is split into two subspaces, where one subspace is the standard finite element space associated to the background mesh and the second subspace is spanned by all cut basis functions corresponding to nodes on the cut elements. We will show that this splitting is stable, uniformly in the discretization parameter and in the location of the interface in the triangulation. Based on this we introduce an efficient preconditioner that is uniformly spectrally equivalent to the stiffness matrix. Using a similar splitting, it is shown that the same preconditioning approach can also be applied to a fictitious domain CutFEM discretization of the Poisson equation. Results of numerical experiments are included that illustrate optimality of such preconditioners for the Poisson interface problem and the Poisson fictitious domain problem.     

Analysis of the projected coupled cluster method in electronic structure calculation

The electronic Schrödinger equation plays a fundamental role in molcular physics. It describes the stationary nonrelativistic behaviour of an quantum mechanical N electron system in the electric field generated by the nuclei. The (Projected) Coupled Cluster Method has been developed for the numerical computation of the ground state energy and wave function. It provides a powerful tool for high accuracy electronic structure calculations. The present paper aims to provide a rigorous analytical treatment and convergence analysis of this method. If the discrete Hartree Fock solution is sufficiently good, the quasi-optimal convergence of the projected coupled cluster solution to the full CI solution is shown. Under reasonable assumptions also the convergence to the exact wave function can be shown in the Sobolev H ¹-norm. The error of the ground state energy computation is estimated by an Aubin Nitsche type approach. Although the Projected Coupled Cluster method is nonvariational it shares advantages with the Galerkin or CI method. In addition it provides size consistency, which is considered as a fundamental property in many particle quantum mechanics.     

Stochastic mathematical programs with hybrid equilibrium constraints

This paper considers a stochastic mathematical program with hybrid equilibrium constraints (SMPHEC), which includes either “here-and-now” or “wait-and-see” type complementarity constraints. An example is given to describe the necessity to study SMPHEC. In order to solve the problem, the sampling average approximation techniques are employed to approximate the expectations and smoothing and penalty techniques are used to deal with the complementarity constraints. Limiting behaviors of the proposed approach are discussed. Preliminary numerical experiments show that the proposed approach is applicable.     

A posteriori error analysis for a boundary element method with nonconforming domain decomposition

We present and analyze an a posteriori error estimator based on mesh refinement for the solution of the hypersingular boundary integral equation governing the Laplacian in three dimensions. The discretization under consideration is a nonconforming domain decomposition method based on the Nitsche technique. Assuming a saturation property, we establish quasireliability and efficiency of the error estimator in comparison with the error in a natural (nonconforming) norm. Numerical experiments with uniform and adaptively refined meshes confirm our theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 947–963, 2014     

Ein Kriterium für die Existenz nicht-linearer ganzer Lösungen elliptischer Differentialgleichungen

Ohne ZusammenfassungDiese Arbeit ist von einem von uns (Johannes Nitsche) unter Contract Nonr — 710(16) zwischen der Universität von Minnesota und dem Office of Naval Research vorrereitet worden.     

A new approach to the analysis of random methods for detecting necessary linear inequality constraints

A new approach is given for the analysis of random methods for detecting necessary constraints in systems of linear inequality constraints. This new approach directly accounts for the fact that two constraints are detected as necessary (hit) at each iteration of a random method. The significance of this two-hit analysis is demonstrated by comparing it with the usual one-hit analysis.     

A smoothing-regularization approach to mathematical programs with vanishing constraints

We consider a numerical approach for the solution of a difficult class of optimization problems called mathematical programs with vanishing constraints. The basic idea is to reformulate the characteristic constraints of the program via a nonsmooth function and to eventually smooth it and regularize the feasible set with the aid of a certain smoothing- and regularization parameter t>0 such that the resulting problem is more tractable and coincides with the original program for t=0. We investigate the convergence behavior of a sequence of stationary points of the smooth and regularized problems by letting t tend to zero. Numerical results illustrating the performance of the approach are given. In particular, a large-scale example from topology optimization of mechanical structures with local stress constraints is investigated.     

Sparsity reconstruction in electrical impedance tomography: An experimental evaluation

We investigate the potential of sparsity constraints in the electrical impedance tomography (EIT) inverse problem of inferring the distributed conductivity based on boundary potential measurements. In sparsity reconstruction, inhomogeneities of the conductivity are a priori assumed to be sparse with respect to a certain basis. This prior information is incorporated into a Tikhonov-type functional by including a sparsity-promoting ?¹-penalty term. The functional is minimized with an iterative soft shrinkage-type algorithm. In this paper, the feasibility of the sparsity reconstruction approach is evaluated by experimental data from water tank measurements. The reconstructions are computed both with sparsity constraints and with a more conventional smoothness regularization approach. The results verify that the adoption of ?¹-type constraints can enhance the quality of EIT reconstructions: in most of the test cases the reconstructions with sparsity constraints are both qualitatively and quantitatively more feasible than that with the smoothness constraint.     

A stable approach to Newton's method for general mathematical programming problems inR n

The usual approach to Newton's method for mathematical programming problems with equality constraints leads to the solution of linear systems ofn +m equations inn +m unknowns, wheren is the dimension of the space andm is the number of constraints. Moreover, these linear systems are never positive definite. It is our feeling that this approach is somewhat artificial, since in the unconstrained case the linear systems are very often positive definite. With this in mind, we present an alternate Newton-like approach for the constrained problem in which all the linear systems are of order less than or equal ton. Furthermore, when the Hessian of the Lagrangian at the solution is positive definite (a situation frequently occurring), all our systems will be positive definite. Hence, in all cases, our Newton-like method offers greater numerical stability. We demonstrate that the convergence properties of this Newton-like method are superior to those of the standard approach to Newton's method. The operation count for the new method using Gaussian elimination is of the same order as the operation count for the standard method. However, if the Hessian of the Lagrangian at the solution is positive definite and we use Cholesky decomposition, then the order of the operation count for the new method is half that for the standard approach to Newton's method. This theory is generalized to problems with both equality and inequality constraints.     

A matheuristic approach with nonlinear subproblems for large-scale packing of ellipsoids

The problem of packing ellipsoids in the three-dimensional space is considered in the present work. The proposed approach combines heuristic techniques with the resolution of recently introduced nonlinear programming models in order to construct solutions with a large number of ellipsoids. The introduced approach is able to pack identical and non-identical ellipsoids within a variety of containers. Moreover, it allows the inclusion of additional positioning constraints. This fact makes the proposed approach suitable for constructing large-scale solutions with specific positioning constraints in which density may not be the main issue. Numerical experiments illustrate that the introduced approach delivers good quality solutions with a computational cost that scales linearly with the number of ellipsoids; and solutions with more than a million ellipsoids are exhibited.