Convergence analysis of Schwarz methods without overlap for the Helmholtz equation

In this paper, the continuous and discrete optimal transmission conditions for the Schwarz algorithm without overlap for the Helmholtz equation are studied. Since such transmission conditions lead to non-local operators, they are approximated through two different approaches. The first approach, called optimized, consists of an approximation of the optimal continuous transmission conditions with partial differential operators, which are then optimized for efficiency. The second approach, called approximated, is based on pure algebraic operations performed on the optimal discrete transmission conditions. After demonstrating the optimal convergence properties of the Schwarz algorithm new numerical investigations are performed on a wide range of unstructured meshes and arbitrary mesh partitioning with cross points. Numerical results illustrate for the first time the effectiveness, robustness and comparative performance of the optimized and approximated Schwarz methods on a model problem and on industrial problems.     

Discrete maximum principle for finite element parabolic models in higher dimensions

When we construct continuous and/or discrete mathematical models in order to describe a real-life problem, these models should have various qualitative properties, which typically arise from some basic principles of the modelled phenomena. In this paper we investigate this question for the numerical solution of initial-boundary problems for the parabolic problems in higher dimensions, with the first boundary condition, using the linear finite elements. We give the conditions for the geometry of the mesh and for the choice of the discretization parameters, i.e., for the step sizes under which the discrete qualitative properties hold. For the special regular uniform simplicial mesh we define the conditions for the discretization step-sizes.     

Adaptive Finite Element Approximation for an Elliptic Optimal Control Problem with Both Pointwise and Integral Control Constraints

In this paper, we study the adaptive finite element approximation for a constrained optimal control problem with both pointwise and integral control constraints. We first obtain the explicit solutions for the variational inequalities both in the continuous and discrete cases. Then a priori error estimates are established, and furthermore equivalent a posteriori error estimators are derived for both the state and the control approximation, which can be used to guide the mesh refinement for an adaptive multi-mesh finite element scheme. The a posteriori error estimators are implemented and tested with promising numerical results.     

PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes

We present an efficient Matlab code for structural topology optimization that includes a general finite element routine based on isoparametric polygonal elements which can be viewed as the extension of linear triangles and bilinear quads. The code also features a modular structure in which the analysis routine and the optimization algorithm are separated from the specific choice of topology optimization formulation. Within this framework, the finite element and sensitivity analysis routines contain no information related to the formulation and thus can be extended, developed and modified independently. We address issues pertaining to the use of unstructured meshes and arbitrary design domains in topology optimization that have received little attention in the literature. Also, as part of our examination of the topology optimization problem, we review the various steps taken in casting the optimal shape problem as a sizing optimization problem. This endeavor allows us to isolate the finite element and geometric analysis parameters and how they are related to the design variables of the discrete optimization problem. The Matlab code is explained in detail and numerical examples are presented to illustrate the capabilities of the code.     

A parallel polynomial Jacobi–Davidson approach for dissipative acoustic eigenvalue problems

We consider a rational algebraic large sparse eigenvalue problem arising in the discretization of the finite element method for the dissipative acoustic model in the pressure formulation. The presence of nonlinearity due to the frequency-dependent impedance poses a challenge in developing an efficient numerical algorithm for solving such eigenvalue problems. In this article, we reformulate the rational eigenvalue problem as a cubic eigenvalue problem and then solve the resulting cubic eigenvalue problem by a parallel restricted additive Schwarz preconditioned Jacobi–Davidson algorithm (ASPJD). To validate the ASPJD-based eigensolver, we numerically demonstrate the optimal convergence rate of our discretization scheme and show that ASPJD converges successfully to all target eigenvalues. The extraneous root introduced by the problem reformulation does not cause any observed side effect that produces an undesirable oscillatory convergence behavior. By performing intensive numerical experiments, we identify an efficient correction-equation solver, an effective algorithmic parameter setting, and an optimal mesh partitioning. Furthermore, the numerical results suggest that the ASPJD-based eigensolver with an optimal mesh partitioning results in superlinear scalability on a distributed and parallel computing cluster scaling up to 192 processors.     

Combinatorial Construction of Seamless Parameter Domains

The problem of seamless parametrization of surfaces is of interest in the context of structured quadrilateral mesh generation and spline-based surface approximation. It has been tackled by a variety of approaches, commonly relying on continuous numerical optimization to ultimately obtain suitable parameter domains. We present a general combinatorial seamless parameter domain construction, free from the potential numerical issues inherent to continuous optimization techniques in practice. The domains are constructed as abstract polygonal complexes which can be embedded in a discrete planar grid space, as unions of unit squares. We ensure that the domain structure matches any prescribed parametrization singularities (cones) and satisfies seamlessness conditions. Surfaces of arbitrary genus are supported. Once a domain suitable for a given surface is constructed, a seamless and locally injective parametrization over this domain can be obtained using existing planar disk mapping techniques, making recourse to Tutte's classical embedding theorem.     

Adaptive Wavelet Methods on Unbounded Domains

In this paper, we introduce an adaptive wavelet method for operator equations on unbounded domains. We use wavelet bases on ? ⁿ to equivalently express the operator equation in terms of a well-conditioned discrete problem on sequence spaces. By realizing an approximate adaptive operator application also for unbounded domains, we obtain a scheme that is convergent at an asymptotically optimal rate. We show the quantitative performance of the scheme by various numerical experiments.     

Optimal-observer design for linear dynamical systems with uncertain parameters

This paper presents a design technique for the synthesis of robust observers for linear dynamical systems with uncertain parameters. The perturbations under consideration are modelled as unknown but bounded disturbances and an ellipsoidal set-theoretic approach is used to formulate the optimal-observer design problem. The optimal criterion introduced here is the minimization of the ‘size’ of the bounding ellipsoid of the estimation error. A necessary and sufficient condition for this optimal design problem is presented. The results are stated in terms of a reduced-order observer with constant gain matrix, which is then determined by solving a matrix Riccati-type equation. Furthermore, a gradient-search algorithm is presented to find the optimal solution when the free parameter that enters in the construction of the bounding ellipsoids of the estimation error is considered as a design parameter. The effectiveness of the proposed approach is illustrated through a numerical example.     

Penalty-Methode bei der numerischen Behandlung von quasilinearen elliptischen Differentialgleichungen vierter und höherer Ordnung

In the following paper we treat the numerical solution of quasilinear elliptic differential equations of fourth and higher order which are Euler-equations of certain variational problems We reduce the differential equation to a system of equations of the second order and solve this system by the method of finite differences. Existence and uniqueness of a minimal solution of the discrete problem and convergence to the solution of the variational problem under the assumptions of consistency and stability are established as the mesh size and the Penalty-parameter tend to zero.     

一种改进的基于深度神经网络的偏微分方程求解方法

偏微分方程求解是计算流体力学等科学与工程领域中数值分析的计算核心。由于物理的多尺度特性和对离散网格质量的敏感性,传统的数值求解方法通常包含复杂的人机交互和昂贵的网格剖分开销,限制了其在许多实时模拟和优化设计问题上的应用效率。提出了一种改进的基于深度神经网络的偏微分方程求解方法TaylorPINN。该方法利用深度神经网络的万能逼近定理和泰勒公式的函数拟合能力,实现了无网格的数值求解过程。在Helmholtz、Klein-Gordon和Navier-Stokes方程上的数值实验结果表明,TaylorPINN能够很好地拟合计算域内时空点坐标与待求函数值之间的映射关系,并提供了准确的数值预测结果。与常用的基于物理信息神经网络方法相比,对于不同的数值问题,TaylorPINN将预测精度提升了3~20倍。     

A Galerkin finite element method for time-fractional stochastic heat equation

In this study, a Galerkin finite element method is presented for time-fractional stochastic heat equation driven by multiplicative noise, which arises from the consideration of heat transport in porous media with thermal memory with random effects. The spatial and temporal regularity properties of mild solution to the given problem under certain sufficient conditions are obtained. Numerical techniques are developed by the standard Galerkin finite element method in spatial direction, and Gorenflo–Mainardi–Moretti–Paradisi scheme is applied in temporal direction. The convergence error estimates for both semi-discrete and fully discrete schemes are established. Finally, numerical example is provided to verify the theoretical results.     

Efficient optimal design of uncertain discrete time dynamical systems

In this paper we consider the problem of optimal design of an uncertain discrete time nonlinear dynamical system. The problem is formulated using an a-posterori design criterion, which can account for uncertainties generated by the dynamics of the system itself as well as parametric uncertainties. In general, for most uncertain complex dynamical systems, this type of method is difficult to solve analytically. A numerical scheme is developed for the optimal design that involves two steps. First, in order to obtain a numerical algorithm for the optimal solution, we apply randomized algorithms for average performance synthesis to approximate the optimal solution. Second, using the properties of the Perron–Frobenius operator we develop an efficient computation approach to calculate the stationary distribution for the uncertain dynamical systems and the average performance criteria.     

Truss topology optimization with discrete design variables—Guaranteed global optimality and benchmark examples

This paper considers the problem of optimal truss topology design subject to multiple loading conditions. We minimize a weighted average of the compliances subject to a volume constraint. Based on the ground structure approach, the cross-sectional areas are chosen as the design variables. While this problem is well-studied for continuous bar areas, we consider in this study the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single available bar area, i.e., a 0/1 problem. In contrast to the heuristic methods considered in many other approaches, our goal is to compute guaranteed globally optimal structures. This is done by a branch-and-bound method for which convergence can be proven. In this branch-and-bound framework, lower bounds of the optimal objective function values are calculated by treating a sequence of continuous but non-convex relaxations of the original mixed-integer problem. The main effect of using this approach lies in the fact that these relaxed problems can be equivalently reformulated as convex problems and, thus, can be solved to global optimality. In addition, these convex problems can be further relaxed to quadratic programs for which very efficient numerical solution procedures exist. By exploiting this special problem structure, much larger problem instances can be solved to global optimality compared to similar mixed-integer problems. The main intention of this paper is to provide optimal solutions for single and multiple load benchmark examples, which can be used for testing and validating other methods or heuristics for the treatment of this discrete topology design problem.     

Designing robust structures - A nonlinear simulation based approach

This paper presents a generally applicable numerical procedure for designing robust structures under uncertainty, which can be coupled with any arbitrary nonlinear computational model for statical or dynamic structural analysis. Based on the results from an uncertain structural analysis several permissible design domains are determined with the aid of cluster analysis methods instead of traditionally computing only one particular set of crisp design parameter values; these represent design alternatives. To identify a preference solution, a discrete three-criteria optimization problem is formulated, which is focused on maximum structural robustness and includes a safety component. A measure for the global robustness of the design alternatives is introduced based on an analog to Shannon’s entropy. The goal of the resulting design is that the structural behavior is only marginally affected by uncertainty and by changes in the design parameters, which further provides comfortable decision margins to the construction engineer.The proposed procedure is demonstrated by means of a numerical example and of an example from engineering practice in vehicle crashworthiness design.     

Optimal shapes of mechanically motivated surfaces

Subject of this contribution is form finding of “optimal” structural shapes with regard to the load carrying behaviour of surface structures under certain load cases. In general, those optimal shapes prefer a membrane state of stress to transfer loading. Bending is omitted as much as possible. It will be focused on two different disciplines and related numerical approaches which deal with solutions of the mentioned task: form finding of prestressed membranes and general shape optimization. As design is an inverse problem both approaches share similar problematic properties as e.g. indeterminate in-plane location of surface discretization or necessary regularization and filtering of sensitivity and other data. As it will turn out, those remedies found for the very special methods of membrane design can be abstracted and transferred to general optimization procedures. That merges into elegant, numerical shape optimal design techniques which combine advantages of both approaches and allow for effective and efficient shape optimization of free formed surfaces, directly on the finite element mesh and for a large number of variables. Typical applications are, for example, membrane design, free form architecture and structural engineering, and metal sheet design.     

Absorbing boundary conditions for time harmonic wave propagation in discretized domains

While many successful absorbing boundary conditions (ABCs) are developed to simulate wave propagation into unbounded domains, most of them ignore the effect of interior discretization and result in spurious reflections at the artificial boundary. We tackle this problem by developing ABCs directly for the discretized wave equation. Specifically, we show that the discrete system (mesh) can be stretched in a non-trivial way to preserve the discrete impedance at the interface. Similar to the perfectly matched layers (PML) for continuous wave equation, the stretch is designed to introduce dissipation in the exterior, resulting in a PML-type ABC for discrete media. The paper includes detailed formulation of the new discrete ABC, along with the illustration of its effectiveness over continuous ABCs with the help of error analysis and numerical experiments. For time-harmonic problems, the improvement over continuous ABCs is achieved without any computational overhead, leading to the conclusion that the discrete ABCs should be used in lieu of continuous ABCs.     

Best domain for an elliptic problem in cartesian coordinates by means of shape‐measure

In (ZAA J. Anal. Appl., Vol. 16, No. 1, pp. 143–155) we introduced a method to determine the optimal domains for elliptic optimal‐shape design problems in polar coordinates. However, the same problem in cartesian coordinates, which are more applicable, is found to be much harder, therefore we had to develop a new approach for these designs. Herein, the unknown domain is divided into a fixed and a variable part and the optimal pair of the domain and its optimal control, is characterized in two stages. Firstly, the optimal control for the each given domain is determined by changing the problem into a measure‐theoretical one, replacing this with an infinite dimensional linear programming problem and approximating schemes; then the nearly optimal control function is characterized. Therefore a function that offers the optimal value of the objective function for a given domain, is defined. In the second stage, by applying a standard optimization method, the global minimizer pair will be obtained. Some numerical examples are also given. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Optimal design of infinite repetitive structures

An approach for designing optimal repetitive structures under arbitrary static loading is presented. It is shown that the analysis of such infinite structures can be reduced to the analysis of the repeating module under transformed loading and boundary conditions. Consequently, both the design parameters and the analysis variables constitute a relatively small set which facilitates the optimization process. The approach hinges on the representative cell method. It is based on formulating the analysis equations and the continuity conditions for a sequence of typical modules. Then, by means of the discrete Fourier transform this problem translates into a boundary value problem of a representative cell in transformed variables, which can be solved by any appropriate analytical or numerical method. The real structural response any-where in the structure is then obtained by the inverse transform. The sensitivities can also be calculated on the basis of the sensitivities of the representative cell. The method is illustrated by the design for minimum compliance with a volume constraint of an infinite plane truss. It is shown that by employing this analysis method within an optimal design scheme one can incorporate a reduced analysis problem in an intrinsically small design space.     

An integro-differential generalization and dynamically consistent discretizations of some hyperbolic models with nonlinear damping

This note presents a weak generalization of a time-delayed partial differential equation which, in turn, generalizes the well-known Burger–Fisher and Burgers–Huxley models. In this work, we provide a full discretization which is consistent with the integro-differential equation under consideration. The main analytical result of this note establishes that the discrete temporal rate of change of the discretization yields a consistent approximation to the differential form of the integro-differential equation investigated. Some numerical examples are provided in order to assess the efficiency and effectiveness of our methodology.