二维稳态热传导问题的正六边形流形元研究

发展了用于分析二维稳态热传导问题的多边形数值流形方法（numerical manifold method,NMM）.根据热传导问题的控制方程、边界条件以及多边形NMM的温度近似函数,采用修正变分原理导出了多边形NMM求解稳态热传导问题的总体方程,给出了多边形单元上的域积分策略.考虑到NMM中数学覆盖系统可不与物理域边界一致以及规则单元的精度优势,采用Wachspress正六边形数学单元对两个典型热传导问题进行了仿真,计算结果与参考解能较好地吻合,表明多边形NMM可以很好地模拟平面稳态热传导问题.     

A modified numerical manifold method for simulation of finite deformation problem

With many people contributing to its modifications and advancements, the numerical manifold method (NMM) is now recognized as an efficient tool to solve the continuum–discontinuum coupling problem in geotechnical engineering. However, false solutions have been found when modeling finite deformation problems using the original NMM. Based on the finite deformation theory, a modified version of NMM is derived from the weak form of conservation of momentum and the corresponding traction boundary condition. By taking the dual cover system as the displacement approximation, the governing equations of the modified NMM are formulated. A comparison of the governing equations of the original NMM and modified NMM illustrates the reason that the original NMM is not suitable for simulation of finite deformation problems. Three numerical examples are investigated to verify the capability of proposed method to predict static and dynamic finite deformation response. Numerical results show that the modified NMM eliminates the errors caused by large rotation and large strain, and obtains a good agreement with analytical solutions and the finite element method.     

Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations

Some efficient and accurate algorithms based on the ultraspherical-Galerkin method are developed and implemented for solving 2nth-order linear differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a spectral discretization. We extend the proposed algorithms to solve the two-dimensional 2nth-order differential equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost and roundoff errors.     

Reuse,Recycle, Reduce (3R) – strategies for the calculation of transient magnetic fields

The discretization of transient magneto-dynamic field problems with geometric discretization schemes such as the Finite Integration Technique or the Finite-Element Method based on Whitney form functions results in nonlinear differential-algebraic systems of equations of index 1. Their time integration with embedded s-stage singly diagonal implicit Runge–Kutta methods requires the solution of s nonlinear systems within one time step. Accelerated solution of these schemes is achieved with techniques following so-called 3R-strategies (“reuse, recycle, reduce”). This involves e.g. the solution of the linear(-ized) equations in each time step where the solution process of the iterative preconditioned conjugate gradient method reuses and recycles spectral information of linear systems from previous stages. Additionally, a combination of an error controlled spatial adaptivity and an error controlled implicit Runge–Kutta scheme is used to reduce the number of unknowns for the algebraic problems effectively and to avoid unnecessary fine grid resolutions both in space and time. First numerical results for 2D nonlinear magneto-dynamic problems validate the presented approach and its implementation. The space discretization in the numerical examples is done by Lagrangian nodal finite elements but the presented algorithms also work in combination with other discretization schemes for the Maxwell equations such as the Whitney vector finite elements.     

Functions orthogonal to polynomials and their application in axially symmetric problems in physics

We investigate properties of sets of functions comprising countably many elements A_n such that every function A_n is orthogonal to all polynomials of degrees less than n. We propose an effective method for solving Fredholm integral equations of the first kind whose kernels are generating functions for these sets of functions. We study integral equations used to solve some axially symmetric problems in physics. We prove that their kernels are generating functions that produce functions in the studied families and find these functions explicitly. This allows determining the elements of the matrices of systems of linear equations related to the integral equations for considering the physical problems.     

A prewavelet-based algorithm for the solution of second-order elliptic differential equations with variable coefficients on sparse grids

We present a Ritz-Galerkin discretization on sparse grids using prewavelets, which allows us to solve elliptic differential equations with variable coefficients for dimensions d ≥ 2. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This algorithm is based on standard 1-dimensional restrictions and prolongations, a simple prewavelet stencil, and the classical operator-dependent stencil for multilinear finite elements. Numerical simulation results are presented for a three-dimensional problem on a curvilinear bounded domain and for a six-dimensional problem with variable coefficients. Simulation results show a convergence of the discretization according to the approximation properties of the finite element space. The condition number of the stiffness matrix can be bounded below 10 using a standard diagonal preconditioner.     

A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations

In this paper, we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three-dimensional non-linear biharmonic partial differential equations on a 19-point compact stencil using coupled approach. The numerical solutions of unknown variable u(x,y,z) and its Laplacian ∇²u are obtained at each internal grid point. The resulting stencil algorithm is presented which can be used to solve many physical problems. The proposed method allows us to use the Dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. The new method is tested on three problems and the results are compared with the corresponding second order approximation, which we also discuss using coupled approach.     

The Optimal Discretization of Stochastic Differential Equations

We study pathwise approximation of scalar stochastic differential equations. The mean squared L₂-error and the expected number n of evaluations of the driving Brownian motion are used for the comparison of arbitrary methods. We introduce an adaptive discretization that reflects the local properties of every single trajectory. The corresponding error tends to zero like c·n^−1/2, where c is the average of the diffusion coefficient in space and time. Our method is justified by the matching lower bound for arbitrary methods that are based on n evaluations on the average. Hence the adaptive discretization is asymptotically optimal. The new method is very easy to implement, and about 7 additional arithmetical operations are needed per evaluation of the Brownian motion. Hereby we can determine the complexity of pathwise approximation of stochastic differential equations. We illustrate the power of our method already for moderate accuracies by means of a simulation experiment.     

Solution to fuzzy system of linear equations with crisp coefficients

This paper presents a new and simple method to solve fuzzy real system of linear equations by solving two n × n crisp systems of linear equations. In an original system, the coefficient matrix is considered as real crisp, whereas an unknown variable vector and right hand side vector are considered as fuzzy. The general system is initially solved by adding and subtracting the left and right bounds of the vectors respectively. Then obtained solutions are used to get a final solution of the original system. The proposed method is used to solve five example problems. The results obtained are also compared with the known solutions and found to be in good agreement with them.     

Torus data flow for parallel computation of missized matrix problems

A data-flow approach is used to solve dense symmetric systems of equations on a torus-connected 2-D mesh of processors. A torus mapping of the matrix onto this processor array allows the Cholesky decomposition to be completed in 3n − 2 time steps using only n²/4 processors (less than half the number needed in previously reported results). New definitions for missized problems and parallel algorithm performance are given along with various time-step, efficiency, and processor utilization plots.     

Solving the path cover problem on circular-arc graphs by using an approximation algorithm

A path cover of a graph G=(V,E) is a family of vertex-disjoint paths that covers all vertices in V. Given a graph G, the path cover problem is to find a path cover of minimum cardinality. This paper presents a simple O(n)-time approximation algorithm for the path cover problem on circular-arc graphs given a set of n arcs with endpoints sorted. The cardinality of the path cover found by the approximation algorithm is at most one more than the optimal one. By using the result, we reduce the path cover problem on circular-arc graphs to the Hamiltonian cycle and Hamiltonian path problems on the same class of graphs in O(n) time. Hence the complexity of the path cover problem on circular-arc graphs is the same as those of the Hamiltonian cycle and Hamiltonian path problems on circular-arc graphs.     

Space-time discontinuous galerkin method for maxwell equations in dispersive media

In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the underlying problem. Unconditional L²-stability and error estimate of order O(τ^r+1+h^k+1/2) are obtained when polynomials of degree at most r and k are used for the temporal discretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r+1 in temporal variable t.     

Heuristic procedures for the m-partial cover problem on a plane

The m-partial cover problem on a plane is defined as follows: given npoints located at cartesian coordinates (x_j, y_j) (j=1,…,n) each with an associated weight w_j, a critical distance R and an integer number m, determine the location of m centres so that the sum of the weights of those points lying within distance R of at least one centre is maximised. Five heuristic procedures to solve the m-partial cover problem are presented and computational experience reported. The use of the procedures for some related problems is discussed.     

CGS algorithms for constrained extremization of functions

CGS (conjugate Gram-Schmidt) algorithms are given for computing extreme points of a real-valued functionf(x) ofn real variables subject tom constraining equationsg(x)=0,M<n. The method of approach is to solve the system $$\begin{gathered} f'(x) + g'(x)*\lambda = 0 \hfill \\ g(x) = 0 \hfill \\ \end{gathered} $$ where λ is the Lagrange multiplier vector, by means of CGS algorithms for systems of nonlinear equations. Results of the algorithms applied to test problems are given, including several problems having inequality constraints treated by adjoining slack variables.     

A high-order numerical manifold method with continuous stress/strain field

Recent attempts to solve solid mechanical problems using the numerical manifold method (NMM) are very fruitful. In the present work, a high-order numerical manifold method (HONMM) which is able to obtain continuous stress/strain field is proposed. By employing the same discretized model as the traditional NMM (TNMM), the proposed HONMM can yield much better accuracy without increasing the number of degrees of freedom (DOFs), and obtain continuous stress/strain field without recourse any stress smoothing operation in the post-processing stage. In addition, the “linear dependence” (LD) issue does not exist in the HONMM, and traditional equation solvers can be employed to solve the simultaneous algebraic equations. A number of numerical examples including four linear elastic continuous problems and five cracked problems are solved with the proposed method. The results show that the proposed HONMM performs much better than the TNMM.     

Mathematical and numerical modellization of large‐scale oceanic waves

This paper is devoted to the theoretical and numerical study of a method which computes the variability of current and density in an oceanic domain. The equations are of Navier–Stokes type for the velocity and of transport‐diffusion type for the density. They are linearized around a given mean circulation and modified by physical assumptions including hydrostatic approximation. The existence and uniqueness of a solution are proved for two sets of equations: first the three‐dimensional problem and then the two‐dimensional cyclic problem derived by assuming a sinusoïdal x‐dependence for the perturbation of the mean flow. The latter corresponds to a modellization of tropical instability waves which are illustrated by the ‘El Nino’ phenomenon. These two problems differ from classical ones because of hydrostatic approximation, boundary conditions imposed by the oceanic domain and complex‐valued functions for the cyclic case. A numerical model is developed for the two‐dimensional cyclic equations. Time discretization is performed by the characteristics method; space discretization uses Q₁ finite elements. Numerical results are presented in a realistic case corresponding to the tropical Pacific Ocean. Copyright © 1999 John Wiley & Sons. Ltd.     

Solving Linear Systems Whose Elements Are Nonlinear Functions of Intervals

The paper addresses the problem of solving linear algebraic systems the elements of which are, in the general case, nonlinear functions of a given set of independent parameters taking on their values within prescribed intervals. Three kinds of solutions are considered: (i) outer solution, (ii) interval hull solution, and (iii) inner solution. A simple direct method for computing a tight outer solution to such systems is suggested. It reduces, essentially, to inverting a real matrix and solving a system of real linear equations whose size n is the size of the original system. The interval hull solution (which is a NP-hard problem) can be easily determined if certain monotonicity conditions are fulfilled. The resulting method involves solving n+1 interval outer solution problems as well as 2n real linear systems of size n. A simple iterative method for computing an inner solution is also given. A numerical example illustrating the applicability of the methods suggested is solved.     

Linear equations and interpolation in Boolean algebra

The authors generalize the classical interpolation formula for Boolean functions of n variables. A characterization of all interpolating systems with 2ⁿ elements is obtained. The methods of proof used are intimately related to the solution of linear Boolean equations.     

AN ITERATIVE SUBSTRUCTURING METHOD WITH NONCONFORMING ELEMENTS

When the red-black subdivisions are satisfied, an iterative substructuring method is proposed to solve the algebraic system of equations arising from the discretization of symmetric elliptic problems via nonconforming finite elements which are only continuous at the quasi-uniform mesh nodes. Theoretical analysis is given and the results of numerical experiments are reported.     

n-sided polygonal hybrid finite elements with unified fundamental solution kernels for topology optimization

In topology optimization, the optimized design can be obtained based on spatial discretization of design domain using natural polygonal finite elements to reduce the influence of mesh geometry on topology optimization solutions. However, the natural polygonal finite elements require separate interpolants for each type of elements and involve troublesome domain integrals. In this study, an alternative n-sided polygonal hybrid finite element possessing multiple-node connection is formulated in a unified form to compress the checkerboard patterns caused by numerical instability in topology optimization. Different from the natural polygonal finite elements, the present polygonal hybrid finite elements involve two sets of independent displacement fields. The intra-element displacement field defined inside the element is approximated by the linear combination of the fundamental solution of the problem to achieve the purpose of the local satisfaction of the governing equations of the problem, but not the specific boundary conditions and the inter-element continuity conditions. To overcome such drawback, the inter-element displacement field defined over the entire element boundary is independently approximated by means of the conventional shape function interpolation. As a result, only line integrals along the element boundary are involved in the computation, whose dimension is reduced by one compared to the domain integrals in the natural polygonal finite elements, and more importantly, allowing us to flexibly construct any polygons from Voronoi tessellations in discretizing complex design domains using same fundamental solution kernels. Numerical results obtained indicate that the present n-sided polygonal hybrid finite elements can produce more accurate displacement solutions and smaller mean compliance, compared to the standard finite elements and the natural polygonal finite elements.