A new approach based on embedding Green’s functions into fixed-point iterations for highly accurate solution to Troesch’s problem

In this paper, a novel approach is introduced for the solution of the non-linear Troesch’s boundary value problem. The underlying strategy is based on Green’s functions and fixed-point iterations, including Picard’s and Krasnoselskii–Mann’s schemes. The resulting numerical solutions are compared with both the analytical solutions and numerical solutions that exist in the literature. Convergence of the iterative schemes is proved via manipulation of the contraction principle. It is observed that the method handles the boundary layer very efficiently, reduces lengthy calculations, provides rapid convergence, and yields accurate results particularly for large eigenvalues. Indeed, to our knowledge, this is the first time that this problem is solved successfully for very large eigenvalues, actually the rate of convergence increases as the magnitude of the eigenvalues increases.     

A meshless method for solving 1D time-dependent heat source problem

A novel meshless numerical procedure based on the method of fundamental solutions (MFS) and the heat polynomials is proposed for recovering a time-dependent heat source and the boundary data simultaneously in an inverse heat conduction problem (IHCP). We will transform the problem into a homogeneous IHCP and initial value problems for the first-order ordinary differential equation. An improved method of MFS is used to solve the IHCP and a finite difference method is applied for solving the initial value problems. The advantage of applying the proposed meshless numerical scheme is producing the shape functions which provide the important delta function property to ensure that the essential conditions are fulfilled. Numerical experiments for some examples are provided to show the effectiveness of the proposed algorithm.     

Computational algorithms for solving the coefficient inverse problem for parabolic equations

Among inverse problems for partial differential equations, we distinguish coefficient inverse problems, which are associated with the identification of coefficients and/or the right-hand side of an equation using some additional information. When considering time-dependent problems, the identification of the coefficient dependences on space and on time is usually separated into individual problems. In some cases, we have linear inverse problems (e.g. identification problems for the right-hand side of an equation); this situation essentially simplify their study. This work deals with the problem of determining in a multidimensional parabolic equation the lower coefficient that depends on time only. To solve numerically a non-linear inverse problem, linearized approximations in time are constructed using standard finite difference approximations in space. The computational algorithm is based on a special decomposition, where the transition to a new time level is implemented via solving two standard elliptic problems.     

A computational method for inverse free boundary determination problem

Based on the method of fundamental solutions and discrepancy principle for the choice of location for source points, we extend in this paper the application of the computational method to determine an unknown free boundary of a Cauchy problem of parabolic‐type equation from measured Dirichlet and Neumann data with noises. The standard Tikhonov regularization technique with the L‐curve method for an optimal regularized parameter is adopted for solving the resultant highly ill‐conditioned system of linear equations. Both one‐dimensional and two‐dimensional numerical examples are given to verify the efficiency and accuracy of the proposed computational method. Copyright © 2007 John Wiley & Sons, Ltd.     

A perturbation-based numerical method for solving a three-dimensional axisymmetric indentation problem

The three-dimensional axisymmetric problem of the indentation of a thin compressible linear elastic layer bonded to a rigid foundation is considered. Approximate analytical solutions of the problem that incorporate a large portion of the singular deformation gradients near the edge of the indenter are presented. An accurate closed-form expression for the deformation as well as the deformation gradient throughout the layer is provided and its effectiveness in solving the problem numerically is demonstrated. By incorporating the approximate solution into the numerical scheme the accuracy and convergence rate increase dramatically.     

A two-phase computational scheme for solving bang-bang control problems

This paper focuses on numerical methods for solving time-optimal control problems using discrete-valued controls. A numerical Two-Phase Scheme, which combines admissible optimal control problem formulation with enhanced branch-and-bound algorithms, is introduced to efficiently solve bang-bang control problems in the field of engineering. In Phase I, the discrete restrictions are relaxed, and the resulting continuous problem is solved by an existing optimal control solver. The information on switching times obtained in Phase I is then used in Phase II wherein the discrete-valued control problem is solved using the proposed algorithm. Two numerical examples, including a third-order system and the F-8 fighter aircraft control problem, are presented to demonstrate the use of this proposed scheme. Comparing to STC and CPET methods proposed in the literature, the proposed scheme provides a novel method to find a different switching structure with a better minimum time for the F-8 fighter jet control problem.     

Bordering method for solving systems of linear equations generated by the finite element method in the plate bending problem

A combined iteration algorithm based on the bordering and conjugate gradient methods is proposed to solve systems of linear equations generated by the finite element method in the plate bending problem. The numerical results for the analysis of the convergence rate of the iterative process are presented in the solution of model problems using a classical and modified algorithm of the method of conjugate gradients. The possibility of acceleration of the iterative algorithm is shown. __________ Translated from Problemy Prochnosti, No. 4, pp. 137–145, July–August, 2007.     

A sensitivity-based approach to solving the inverse eigenvalue problem for linear structures carrying lumped attachments

This paper presents an efficient algorithm for designing dynamical systems to exhibit a desired spectrum of eigenvalues. Focusing on combined systems of linear structures carrying various lumped element attachments, we apply the assumed-modes method and the implicit function theorem to derive analytical expressions for eigenvalue sensitivities, which are used to efficiently determine the minimal set of structural modifications needed to achieve a set of desired eigenvalues. The proposed algorithm employs an adaptive step size, performs significantly better than existing approaches, and can be easily applied to a broad range of structures. Convergence properties and limitations on achievable eigenvalues are also discussed, and a number of case studies demonstrating the performance of the algorithm in a wide variety of different applications are also included.     

A universal method for structural static reanalysis of topological modifications

This paper presents a universal method, iterative combined approximation (iterative CA) approach, for structural static reanalysis of all types of topological modifications. The proposed procedure is basically an approximate two-step method. First, the newly added degrees of freedom (DOFs) are assumed to be linked to the original DOFs of the modified structure by means of the Guyan reduction so as to obtain the condensed equation. Second, the displacements of the original DOFs of the modified structure are solved by using the iterative CA approach. And the displacements of the newly added DOFs resulting from topological modification can be recovered. Four numerical examples are given to illustrate the applications of the present approach. The results show that the proposed method is effective for structural static reanalysis of all types of the topological modifications and it is easy to implement on a computer. Copyright © 2004 John Wiley & Sons, Ltd.     

An hp‐adaptive finite element method for scattering problems in computational electromagnetics

An hp‐adaptive finite element (FE) approach is presented for a reliable, efficient and accurate solution of 3D electromagnetic scattering problems. The radiation condition in the far field is satisfied automatically by approximation with infinite elements (IE). Near optimal discretizations that can effectively resolve local rapid variations in the scattered field are sought adaptively by mesh refinements blended with graded polynomial enrichments. The p‐enrichments need not be spatially isotropic. The discretization error can be controlled by a self‐adaptive process, which is driven by implicit or explicit a posteriori error estimates. The error may be estimated in the energy norm or in a quantity of interest. A radar cross section (RCS) related linear functional is used in the latter case. Adaptively constructed solutions are compared to pure uniform p approximations. Numerical, highly accurate, and fairly converged solutions for a number of generic problems are given and compared to previously published results. Copyright © 2004 John Wiley & Sons, Ltd.     

Collocation method based on shifted Chebyshev and radial basis functions with symmetric variable shape parameter for solving the parabolic inverse problem

This work introduces a new numerical solution to the inverse parabolic problem with source control parameter that has important applications in large fields of applied science. We expand the approximate solution of the inverse problem in terms of shifted Chebyshev polynomials in time and radial basis functions with symmetric variable shape parameter in space, with unknown coefficients. Unknown coefficient matrix determined using the collocation technique. Sample results show that the proposed method is very accurate. Moreover, the proposed method is compared with two other methods, fourth-order compact difference scheme and method of lines. Finally, we examine the stability of our method for the case where there is additive noise in input data.     

A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.     

A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low‐rank damping

The low‐rank damping term appears commonly in quadratic eigenvalue problems arising from physical simulations. To exploit the low‐rank damping property, we propose a Padé approximate linearization (PAL) algorithm. The advantage of the PAL algorithm is that the dimension of the resulting linear eigenvalue problem is only n + ?m, which is generally substantially smaller than the dimension 2n of the linear eigenvalue problem produced by a direct linearization approach, where n is the dimension of the quadratic eigenvalue problem, and ? and m are the rank of the damping matrix and the order of a Padé approximant, respectively. Numerical examples show that by exploiting the low‐rank damping property, the PAL algorithm runs 33–47% faster than the direct linearization approach for solving modest size quadratic eigenvalue problems. Copyright © 2015 John Wiley & Sons, Ltd.     

An exact method for solving the manufacturing cell formation problem

The cell formation problem is extensively studied in the literature, but very few authors have proposed exact methods. In this paper a linear binary programming formulation is introduced to generate a solution for the cell formation problem. To verify the behaviour of the proposed model, a set of 35 benchmark problems is solved using the branch and cut method implemented in the IBM ILOG CPLEX 10.11 Optimiser. Moreover, these results allow us to validate the quality of the solution generated with heuristic methods proposed in the literature. This experimentation indicates that, for the smaller problems, the best-known solutions are the same as those generated with CPLEX 10.11 Optimiser. These results indicate a fair confidence in the optimality of the best-known solutions generated by the heuristic methods. Furthermore, our approach is the first exact method providing results of this quality.     

Nitsche's method for Helmholtz problems with embedded interfaces

In this work, we use Nitsche's formulation to weakly enforce kinematic constraints at an embedded interface in Helmholtz problems. Allowing embedded interfaces in a mesh provides significant ease for discretization, especially when material interfaces have complex geometries. We provide analytical results that establish the well‐posedness of Helmholtz variational problems and convergence of the corresponding finite element discretizations when Nitsche's method is used to enforce kinematic constraints. As in the analysis of conventional Helmholtz problems, we show that the inf‐sup constant remains positive provided that the Nitsche's stabilization parameter is judiciously chosen. We then apply our formulation to several 2D plane‐wave examples that confirm our analytical findings. Doing so, we demonstrate the asymptotic convergence of the proposed method and show that numerical results are in accordance with the theoretical analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

A parallel implicit/explicit hybrid time domain method for computational electromagnetics

The numerical solution of Maxwell's curl equations in the time domain is achieved by combining an unstructured mesh finite element algorithm with a cartesian finite difference method. The practical problem area selected to illustrate the application of the approach is the simulation of three‐dimensional electromagnetic wave scattering. The scattering obstacle and the free space region immediately adjacent to it are discretized using an unstructured mesh of linear tetrahedral elements. The remainder of the computational domain is filled with a regular cartesian mesh. These two meshes are overlapped to create a hybrid mesh for the numerical solution. On the cartesian mesh, an explicit finite difference method is adopted and an implicit/explicit finite element formulation is employed on the unstructured mesh. This approach ensures that computational efficiency is maintained if, for any reason, the generated unstructured mesh contains elements of a size much smaller than that required for accurate wave propagation. A perfectly matched layer is added at the artificial far field boundary, created by the truncation of the physical domain prior to the numerical solution. The complete solution approach is parallelized, to enable large‐scale simulations to be effectively performed. Examples are included to demonstrate the numerical performance that can be achieved. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical method for unilateral problems

Variational inequalities connected with Signorini's problem have appeared as a natural generalization of the minimum potential-energy theorem for bodies with unilateral constraints. In this paper, we describe numerical experience on the use of variational inequalities and Pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity, like those describing the equilibrium configuration of an elastic membrane stretched over an elastic obstacle. These problems have the peculiar feature of being alternatively formulated as nonlinear boundary value problems without constraints for which the technique of Pade approximants can be successfully employed. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution.     

A heuristic for solving the redundancy allocation problem for multi-state series-parallel systems

The redundancy allocation problem is formulated with the objective of minimizing design cost, when the system exhibits a multi-state reliability behavior, given system-level performance constraints. When the multi-state nature of the system is considered, traditional solution methodologies are no longer valid. This study considers a multi-state series-parallel system (MSPS) with capacitated binary components that can provide different multi-state system performance levels. The different demand levels, which must be supplied during the system-operating period, result in the multi-state nature of the system. The new solution methodology offers several distinct benefits compared to traditional formulations of the MSPS redundancy allocation problem. For some systems, recognizing that different component versions yield different system performance is critical so that the overall system reliability estimation and associated design models the true system reliability behavior more realistically. The MSPS design problem, solved in this study, has been previously analyzed using genetic algorithms (GAs) and the universal generating function. The specific problem being addressed is one where there are multiple component choices, but once a component selection is made, only the same component type can be used to provide redundancy. This is the first time that the MSPS design problem has been addressed without using GAs. The heuristic offers more efficient and straightforward analyses. Solutions to three different problem types are obtained illustrating the simplicity and ease of application of the heuristic without compromising the intended optimization needs.     

基于有限元法的正交各向异性复合材料结构材料参数识别

以大型商用有限元软件ABAQUS为计算平台,提出了正交各向异性复合材料结构材料参数的识别方法。将材料参数识别的问题转化为极小化目标函数的问题,其中目标函数定义为测量位移与有限元计算的相应位移之差的平方和。采用Levenberg-Marquardt方法极小化目标函数,其中灵敏度的计算基于复合材料的有限元离散结构的求解方程对识别的材料参数求导。数值算例表明本文中提出的方法是有效的。在识别参数过程中,参数的初值以及搜索范围的确定对于识别结果有着重要影响。因此必须充分利用材料参数的先验信息。ABAQUS是高效可靠的商用有限元软件,提出的参数识别方法基于这类商用软件,因而该方法有很强的实用性。     

求解两物种小系统发育问题的遗传算法题

基于复制-丢失比对(DLA)问题模型,研究了复制-丢失(D-L)演化模型下两物种(2-species)小系统发育问题(SPP),缩写为2-SPP-DL问题。通过引入比对算法、标记算法及3种智能变异算子,提出了求解2-SPP-DL问题的遗传算法——G2SP算法。G2SP算法采用普通算子和智能算子相结合的方式,普通算子能有效地保持种群的多样性,而智能算子则能提高种群的收敛性,使其更快地进化到最优解区域。利用4种真实菌属的tRNA和rRNA基因数据对算法性能进行测试,实验结果表明,G2SP算法能够获得较PBLP算法更小的进化代价,是求解2-SPP-DL问题的一种有效方法。