Using integral equations and the immersed interface method to solve immersed boundary problems with stiff forces

We propose a fast, explicit numerical method for computing approximations for the immersed boundary problem in which the boundaries that separate the fluid into two regions are stiff. In the numerical computations of such problems, one frequently has to contend with numerical instability, as the stiff immersed boundaries exert large forces on the local fluid. When the boundary forces are treated explicitly, prohibitively small time-steps may be required to maintain numerical stability. On the other hand, when the boundary forces are treated implicitly, the restriction on the time-step size is reduced, but the solution of a large system of coupled non-linear equations may be required. In this work, we develop an efficient method that combines an integral equation approach with the immersed interface method. The present method treats the boundary forces explicitly. To reduce computational costs, the method uses an operator-splitting approach: large time-steps are used to update the non-stiff advection terms, and smaller substeps are used to advance the stiff boundary. At each substep, an integral equation is computed to yield fluid velocity local to the boundary; those velocity values are then used to update the boundary configuration. Fluid variables are computed over the entire domain, using the immersed interface method, only at the end of the large advection time-steps. Numerical results suggest that the present method compares favorably with an implementation of the immersed interface method that employs an explicit time-stepping and no fractional stepping.     

Shape sensitivity analysis of natural frequencies of structures using boundary elements

The boundary element approach to shape sensitivity analysis of eigenvalues of free vibrating elastic structures is presented. The eigenvalue problem is described in terms of the boundary integral equation method. Using the variational approach for variable regions, first-order sensitivities of simple frequencies are derived. Dependence of eigenvalues with respect to the stochastic shape of the boundary is considered. The numerical procedure of discretization of the problem is characterized. Numerical examples for two-dimensional problems are presented.     

A fast boundary element method for the two-dimensional Helmholtz Equation

In this paper, we present a fast method for solving boundary integral equations arising from the exterior Dirichlet problem for the two-dimensional Helmholtz equation. This method combines a quadrature method for discretizing the boundary integral equations with a preconditioned iterative method for solving the resulting dense, nonsymmetric linear systems. Using this method, a polynomial rate of convergence can be obtained by performing a finite number of iterations, which yields high computational efficiency. Various numerical examples are presented.     

A symmetric variational finite element-boundary integral equation coupling method

This paper is concerned with the development of a mixed variational principle for coupling finite element and boundary integral methods in interface problems, using the generalized Poisson's equation as a prototype situation. One of its primary objectives is to compare the performance of fully variational procedures with methods that use collocation for the treatment of boundary integral equations. A distinctive feature of the new variational principle is that the discretized algebraic equations for the coupled problem are automatically symmetric since they are all derived from a single functional. In addition, the condition that the flux remain continuous across interfaces is satisfied naturally. In discretizing the problem within inhomogeneous or loaded regions, domain finite elements are used to approximate the field variable. On the other hand, only boundary elements are used for regions where the medium is homogeneous and free of external agents. The corresponding integral equations are discretized both by fully variational and by collocation techniques. Results of numerical experiments indicate that the accuracy of the fully variational procedure is significantly greater than that of collocation for the complete interface problem, especially for complex disturbances, at little additional computational cost. This suggests that fully variational procedures may be preferable to collocation, not only in dealing with interface problems, but even for solving integral equations by themselves.     

A Numerical Method for Solving Stochastic Optimal Control Problems with Linear Control

We introduce a numerical method to solve stochastic optimal control problems which are linear in the control. We facilitate the idea of solving two-point boundary value problems with spline functions in order to solve the resulting dynamic programming equation. We then show how to effectively reduce the dimension in the proposed algorithm, which improves computational time and memory constraints. An example, motivated as an invest problem with uncertain cost, is provided, and the effectiveness of our method demonstrated.     

A boundary value approach for a class of linear singularly perturbed boundary value problems

A new boundary value technique, which is simple to use and easy to implement, is presented for a class of linear singularly perturbed two-point boundary value problems with a boundary layer at one end (left or right) point of the underlying interval. As with other methods, the original problem is partitioned into inner and outer solution of differential equations. The method is distinguished by the following fact: the inner region problem is solved as a two-point boundary layer correction problem and the outer region problem of the differential equation is solved as initial-value problem with initial condition at end point. Some numerical experiments have been included to demonstrate the applicability of the proposed method.     

Fast Fourier-Galerkin Methods for Nonlinear Boundary Integral Equations

We develop in this paper a fast Fourier-Galerkin method for solving the nonlinear integral equation which is reformulated from a class of nonlinear boundary value problems. By projecting the nonlinear term onto the approximation subspaces, we make the Fourier-Galerkin method more efficient for solving the nonlinear integral equations. A fast algorithm for solving the resulting discrete nonlinear system is designed by integrating together the techniques of matrix compressing, numerical quadrature for oscillatory integrals, and the multilevel augmentation method. We prove that the proposed method enjoys an optimal convergence order and a nearly linear computational complexity. Numerical experiments are presented to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed method.     

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations because they are flexible with respect to the geometry of the computational domain, they can provide high order convergence, they are not more complicated for problems with many space dimensions and they allow for local refinement. The aim of this paper is to show that the solution of the Rosenau equation, as an example of an initial-boundary value problem with multiple boundary conditions, can be implemented using RBF approximation methods. We extend the fictitious point method and the resampling method to work in combination with an RBF collocation method. Both approaches are implemented in one and two space dimensions. The accuracy of the RBF fictitious point method is analyzed partly theoretically and partly numerically. The error estimates indicate that a high order of convergence can be achieved for the Rosenau equation. The numerical experiments show that both methods perform well. In the one-dimensional case, the accuracy of the RBF approaches is compared with that of the corresponding pseudospectral methods, showing similar or slightly better accuracy for the RBF methods. In the two-dimensional case, the Rosenau problem is solved both in a square domain and in an irregular domain with smooth boundary, to illustrate the capability of the RBF-based methods to handle irregular geometries.     

A Green-function-based multiscale method for uncertainty quantification of finite body random heterogeneous materials

Classical continuum theories are formulated based on the assumption of large scale separation. For scale-coupling problems involving uncertainties, novel multiscale methods are desired. In this study, by employing the generalized variational principles, a Green-function-based multiscale method is formulated to decompose a boundary value problem with random microstructure into a slow scale deterministic problem and a fast scale stochastic one. The slow scale problem corresponds to common engineering practices by smearing out fine-scale microstructures. The fast scale problem evaluates fluctuations due to random microstructures, which is important for scale-coupling systems and particularly failure problems. Two numerical examples are provided at the end.     

Fast numerical upscaling of heat equation for fibrous materials

We are interested in numerical methods for computing the effective heat conductivities of fibrous insulation materials, such as glass or mineral wool, characterized by low solid volume fractions and high contrasts, i.e., high ratios between the thermal conductivities of the fibers and the surrounding air. We consider a fast numerical method for solving some auxiliary cell problems appearing in this upscaling procedure. The auxiliary problems are boundary value problems of the steady-state heat equation in a representative elementary volume occupied by fibers and air. We make a simplification by replacing these problems with appropriate boundary value problems in the domain occupied by the fibers only. Finally, the obtained problems are further simplified by taking advantage of the slender shape of the fibers and assuming that they form a network. A discretization on the graph defined by the fibers is presented and error estimates are provided. The resulting algorithm is discussed and the accuracy and the performance of the method are illusrated on a number of numerical experiments.     

High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Ditkowski (ICASE Report No. 96-8, 1996; and J. Comput. Phys. 133(2), 1997) and Ditkowski (Ph.D. thesis, Tel Aviv University, 1997), for initial boundary value problems on constant irregular domains. We perform a comprehensive theoretical analysis of the numerical issues, which arise when dealing with domains, whose boundaries evolve smoothly in the spatial domain as a function of time. In this class of problems the moving boundaries are impenetrable with either Dirichlet or Neumann boundary conditions, and should not be confused with the class of moving interface problems such as multiple phase flow, solidification, and the Stefan problem. Unlike other similar works on this class of problems, the resulting method is not restricted to domains of up to 3-D, can achieve higher than 2nd-order accuracy both in time and space, and is strictly stable in semi-discrete settings. The strict stability property of the method also implies, that the numerical solution remains consistent and valid for a long integration time. A complete convergence analysis is carried in semi-discrete settings, including a detailed analysis for the implementation of the diffusion equation. Numerical solutions of the diffusion equation, using the method for a 2nd and a 4th-order of accuracy are carried out in one dimension and two dimensions respectively, which demonstrates the efficacy of the method. This research was supported by the Israel Science Foundation (grant No. 1362/04).     

The hierarchial preconditioning on unstructured grids

We present the implementation of two hierarchically preconditioned methods for the fast solution of mesh equations that approximate 2D-elliptic boundary value problems on unstructured quasi uniform triangulations. Based on the fictitious space approach the original problem can be embedded into an auxiliary one, where both the hierarchical grid information and the preconditioner are well defined. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as thebpx-preconditioned cg-iteration having optimal computational costs. Several numerical examples demonstrate the efficiency of the artificially constructed hierarchical methods which can be of importance in industrial engineering, where often only the nodal coordinates and the element connectivity of the underlying (fine) discretization are available.     

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.     

Poisson方程的Q_1~(rot)元和NF_1元特征值法

本文给出用三维非协调元的特征值方法求解一般的二阶椭圆边值问题的数值计算方法,从而验证了非协调元的收敛性的理论正确性及三维Q_1~(rot)元特征值误差渐进展开式的正确性.本文的数值实验表明:三维Q_1~(rot)元外推特征值下逼近准确特征值;三维NF_1元特征值和外推特征值都下逼近准确特征值;三维Q_1~(rot)元和三维NF_1元二网格离散方案特征值既下逼近准确特征值又上逼近准确特征值;三维Q_1~(rot)元比三维NF_1元有较好的数值表现.     

Iterative methods for model reduction by domain decomposition

It is proposed a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the two subdomains. In particular, in one subdomain we discretize the governing equations by a canonical scheme, whereas in the other one we solve a reduced order model of the original problem. Different approaches to couple the low-order model to the usual discretization are presented. The effectiveness of these approaches is tested on numerical examples pertinent to non-linear model problems including Laplace equation with non-linear boundary conditions and compressible Euler equations.     

Singular perturbation method for boundary value problems in two-parameter discrete control systems

The three types of boundary value problem arising in singularly perturbed discrete control systems with two small parameters are considered. Each of these problems possesses three widely different clusters of eigenvalues, resulting in one slow mode and two stable or unstable fast modes and thereby exhibiting a three-time-scale character. Singular perturbation methods are developed to obtain approximate solutions in terms of an outer series solution and two boundary layer correction series solutions corresponding to the two small parameters. Three examples are given to illustrate the proposed methods for the three types of problem.     

Boundary element methods for magnetostatic field problems: a critical view

For the solution of magnetostatic field problems we discuss and compare several boundary integral formulations with respect to their accuracy, their efficiency, and their robustness. We provide fast boundary element methods which are able to deal with multiple connected computational domains, with large magnetic permeabilities, and with complicated structures with small gaps. The numerical comparison is based on several examples, including a controllable reactor as a real-world problem.     

Numerical and computational efficiency of solvers for two-phase problems

We consider two-phase flow problems, modelled by the Cahn–Hilliard equation. In this work, the nonlinear fourth-order equation is decomposed into a system of two coupled second-order equations for the concentration and the chemical potential.We analyse solution methods based on an approximate two-by-two block factorization of the Jacobian of the nonlinear discrete problem. We propose a preconditioning technique that reduces the problem of solving the non-symmetric discrete Cahn–Hilliard system to a problem of solving systems with symmetric positive definite matrices where off-the-shelf multilevel and multigrid algorithms are directly applicable. The resulting solution methods exhibit optimal convergence and computational complexity properties and are suitable for parallel implementation.We illustrate the efficiency of the proposed methods by various numerical experiments, including parallel results for large scale three dimensional problems.