Direct Solution Algorithm Research Articles

An adaptive fast direct solver for boundary integral equations in two dimensions

We describe an algorithm for the rapid direct solution of linear algebraic systems arising from the discretization of boundary integral equations of potential theory in two dimensions. The algorithm is combined with a scheme that adaptively rearranges the parameterization of the boundary in order to minimize the ranks of the off-diagonal blocks in the discretized operator, thus obviating the need for the user to supply a parameterization r of the boundary for which the distance ‖ r ( s ) − r ( t ) ‖ between two points on the boundary is related to their corresponding distance | s − t | in the parameter space. The algorithm has an asymptotic complexity of O ( N log 2 N ) , where N is the number of nodes in the discretization. The performance of the algorithm is illustrated with several numerical examples.

Jacobi spectral Galerkin method for the integrated forms of second-order differential equations

Efficiently direct solvers based on the Jacobi–Galerkin method for the integrated forms of second-order elliptic equations in one and two space variables are presented. They are based on appropriate base functions for the Galerkin formulation which lead to discrete systems with specially structured matrices that can be efficiently inverted. The homogeneous Dirichlet boundary conditions are satisfied exactly by expanding the unknown variable into a polynomial basis of functions which are built upon the Jacobi polynomials. The direct solution algorithm here developed for the homogeneous Dirichlet problem in two-dimensions relies upon a diagonalization process. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.

A More Effective Technique of Design Synthesis for MEMS with Expected Performance

A design synthesis technique based on sensitivity for Micro-Electro-Mechanical Systems (MEMS) proposed. This new technique can be called Sensitivity-Based Direct Solution Algorithm (DSA) of design synthesis for MEMS with expected performance. Design synthesis with expected performance is regarded as a reverse problem of MEMS analysis. Behavior equation group can be educed from analysis equations. Solving the behavior equation group only need L design variables, L is number of desired behaviors. This behavior equation group can be solved using any solution algorithm of non-linear equation group. Newton Iteration Method based on sensitivity is adopted. Comparing with Genetic Optimization Algorithm (GA) and Simulated Annealing Optimization Algorithm (SA), computational workload of DSA is greatly decreased. For instance, synthesis computation of a meandering resonator only needs 4 iterations (17 analyses); computational time is decreased from 7~8 hours to less than 30 seconds.

A Multiderivative Collocation Method for 5th Order Ordinary Differential Equations

Problem statement: The conventional methods of solving higher order differential equations have been by reducing them to systems of first order equations. This approach is cumbersome and increases computational time. Approach: To address this problem, a numerical algorithm for direct solution of 5th order initial value problems in ordinary differential equations (odes), using power series as basis function, is proposed in this research. Collocation of the differential system is taken at selected grid points to reduce the number of functions to be evaluated per iteration. A number of predictors and their derivatives having the same order of accuracy with the main method are proposed. Results: The approach yields a multiderivative method of order six. Numerical examples solved show increased efficiency of the method with increased number of iterations, converging to the theoretical solutions. Conclusion/Recommendations: The new mutiderivative method is efficient to solve linear and nonlinear fifth order odes without reduction to system of lower order equations.

A parallel algorithm for direct solution of large sparse linear systems, well suitable to domain decomposition methods

The goal of this paper is to develop a parallel algorithm for the direct solution of large sparse linear systems and integrate it into domain decomposition methods. The computational effort for these linear systems, often encountered in numerical simulation of structural mechanics problems by finite element codes, is very significant in terms of run-time and memory requirements.In this paper, a two-level parallelism is exploited. The exploitation of the lower level of parallelism is based on the development of a parallel direct solver with a nested dissection algorithm and to introduce it into the FETI methods. This direct solver has the advantage of handling zero-energy modes in floating structures automatically and properly. The upper level of parallelism is a coarse-grain parallelism between substructures of FETI. Some numerical tests are carried out to evaluate the performance of the direct solver.

Solution of indefinite linear systems using an LQ decomposition for the linear constraints

In this paper, indefinite linear systems with linear constraints are considered. We present a special decomposition that makes use of the LQ decomposition, and retains the constraints in the factors. The resulting decomposition is of a structure similar to that obtained using the Bunch–Kaufman–Parlett algorithm. The decomposition can be used in a direct solution algorithm for indefinite systems, but it can also be used to construct effective preconditioners. Combinations of the latter with conjugate gradient type methods have been demonstrated to be very useful.

A parallel algorithm for direct solution of large sparse linear systems, well suitable to domain decomposition methods

The goal of this paper is to develop a parallel algorithm for the direct solution of large sparse linear systems and integrate it into domain decomposition methods. The computational effort for these linear systems, often encountered in numerical simulation of structural mechanics problems by finite element codes, is very significant in terms of run-time and memory requirements.In this paper, a two-level parallelism is exploited. The exploitation of the lower level of parallelism is based on the development of a parallel direct solver with a nested dissection algorithm and to introduce it into the FETI methods. This direct solver has the advantage of handling zero-energy modes in floating structures automatically and properly. The upper level of parallelism is a coarse-grain parallelism between substructures of FETI. Some numerical tests are carried out to evaluate the performance of the direct solver.

Jacobi spectral Galerkin method for elliptic Neumann problems

This paper is concerned with fast spectral-Galerkin Jacobi algorithms for solving one- and two-dimensional elliptic equations with homogeneous and nonhomogeneous Neumann boundary conditions. The paper extends the algorithms proposed by Shen (SIAM J Sci Comput 15:1489–1505, 1994) and Auteri et al. (J Comput Phys 185:427–444, 2003), based on Legendre polynomials, to Jacobi polynomials \(P_{n}^{(\alpha,\beta)}(x)\) with arbitrary α and β. The key to the efficiency of our algorithms is to construct appropriate basis functions with zero slope at the endpoints, which lead to systems with sparse matrices for the discrete variational formulations. The direct solution algorithm developed for the homogeneous Neumann problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Neumann data are accounted for by means of a lifting. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.

Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials

It is well known that for the discretization of the biharmonic operator with spectral methods (Galerkin, tau, or collocation) we have a condition number of O ( N 8 ) , where N is the number of retained modes of approximations. This paper presents some efficient spectral algorithms, for reducing this condition number to O ( N 4 ) , based on the Jacobi–Galerkin methods for fourth-order equations in one variable. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. Jacobi–Galerkin methods for fourth-order equations in two dimension are considered. Numerical results indicate that the direct solvers presented in this paper are significantly more accurate at large N values than that based on the Chebyshev– and Legendre–Galerkin methods.

Efficient optimization of simulated moving bed processes

Simulated moving bed (SMB) chromatography is attracting more and more attention since it is a powerful technique for complex separation tasks. Nowadays, more than 60% of preparative SMB units are installed in the pharmaceutical and in the food industry [SDI, Preparative and Process Liquid Chromatography: The Future of Process Separations, International Strategic Directions, Los Angeles, USA, 2002. http://www.strategic-directions.com]. Chromatography is the method of choice in these fields, because often pharmaceuticals and fine-chemicals have physico-chemical properties which differ little from those of the by-products, and they may be thermally instable [J. Kinkel, M. Schulte, R. Nicoud, F. Charton, Simulated moving bed (SMB) chromatography: an efficient method for performing large-scale separation of optical isomers, Chiral Eur. 95 (1995) 121–132; J. Strube, A. Jupke, A. Epping, H. Schmidt-Traub, M. Schulte, R. Devant, Design, optimization and operation of chromatographic processes in the production of enantiomerically pure pharmaceuticals, Chirality 11 (1999) 440–450]. In these cases, standard separation techniques as distillation and extraction are not applicable [S. Imamoglu, Simulated moving bed chromatography (SMB) for applications in bioseparation, Adv. Biochem. Eng. Biotechnol. 76 (2002) 211–231]. The use of the SMB technology leads to more efficient operations in terms of dilution and adsorbent utilization than the classical batch separation. Optimization of SMB processes is still a challenging task, particularly when a rigorous first-principles process model is used. SMB processes exhibit strong non-linear behavior and they are of periodic nature. Additionally, their hybrid, non-steady and non-ideal characteristics have to be taken into account. Model-based optimization strategies are therefore of great importance for research and practice. In this article, we report new efficient numerical approaches for the solution of the dynamic optimization problem arising from SMB processes [A. Toumi, Optimaler Betrieb und Regelung von Simulated Moving Bed Prozessen, Dissertation, Fachbereich Bio und Chemieingenieurwesen, University of Dortmund, 2004]. They have been developed in the software package MUSCOD-II [D. Leineweber, Efficient reduced SQP methods for the optimization of chemical processes described by large sparse DAE models, Dissertation, vol. 613, VDI Reihe 3, Verfahrenstechnik, VDI Verlag, 1999], a recent implementation of the direct multiple shooting method where the optimal state trajectory and the corresponding operating parameters are determined simultaneously [H. Bock, K. Plitt, A multiple shooting algorithm for direct solution of optimal control problems, in: Proceedings of the 9th IFAC World Congress Budapest, Pergamon Press, 1984, pp. 243–247; H. Bock, Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen, Dissertation, Bonner Mathematische Schriften, 1987, p. 183]. Numerical results show excellent performance for a benchmark enantiomer separation. Finally, the ability to solve the non-linear programs quickly also enables us to consider more challenging operating regimes like VARICOL [P. Adam, R. Nicoud, M. Bailly, O. Ludemann-Hombourger, Process and device for separation with variable-length chromatographic columns, US Patent 6,413,419 (2002)] and PowerFeed [Z. Zhang, M. Mazzotti, M. Morbidelli, PowerFeed operation of simulated moving bed units: changing flow-rates during the switching interval, J. Chromatogr. A 1006 (2003) 87–99].

Object oriented implementation of distributed finite element analysis in .NET

The paper describes a detailed study into the object-oriented implementation of distributed finite element analysis on desktop computers using the .NET framework. The software design aspects are described in some detail for both direct and iterative solution algorithms. The use of interfaces played an important role in the software design. This, together with the .NET framework, enabled remote objects to be implemented in a relatively seamless fashion. The solution routines were “blind” to whether the objects were local or remote. Numerical tests were carried out and reasonable speed-up was achieved, particularly for direct solution methods. It is concluded that .NET provides a viable framework for implementing distributed computing on networks of personal computers.

Efficient spectral-Galerkin algorithms for direct solution of the integrated forms of second-order equations using ultraspherical polynomials

AbstractIt is well known that spectral methods (tau, Galerkin, collocation) have a condition number of O(N4) where N is the number of retained modes of polynomial approximations. This paper presents some efficient spectral algorithms, which have a condition number of O(N2), based on the ultraspherical-Galerkin methods for the integrated forms of second-order elliptic equations in one and two space variables. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. The complexities of the algorithms are a small multiple of Nd+1 operations for a d-dimensional domain with (N – 1)d unknowns, while the convergence rates of the algorithms are exponentials with smooth solutions.

Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials

It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of $O(N^{4})$ ( $N$ is the number of retained modes of polynomial approximations). This paper presents some efficient spectral algorithms, which have a condition number of $O(N^{2})$ , based on the Jacobi–Galerkin methods of second-order elliptic equations in one and two space variables. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. The complexities of the algorithms are a small multiple of $N^{d+1}$ operations for a $d$ -dimensional domain with $(N-1)^d$ unknowns, while the convergence rates of the algorithms are exponentials with smooth solutions.

MATHEMATICAL MODELLING OF ELECTROMAGNETIC SCATTERING FROM A THIN PENETRABLE TARGET

Three mathematical models based on approximate surface integral equations for electromagnetic analysis of scalar wave scattering from thin extended target are considered. Such models include different systems of the second kind singular integral equations determined on the target median. The effective algorithm for direct (without preliminary regularization) numerical solution of the systems is based on the special quadrature formulae for singular integrals. Verification of the mathematical models and their comparison is performed in the case of penetrable cylindrical shell in homogeneous non-magnetic medium.

Efficient Solution of Integral Equations in a Localizing Basis

A sparse direct solution algorithm is reported for discrete representations of boundary integral operators. The algorithm relies on an expansion of the unknown surface currents in a numerically determined basis of functions that are simultaneously localized to small regions on a larger target while also satisfying global boundary conditions. It is shown that the QR factorization of the impedance matrix is sparse in this basis at low to moderate frequencies.

KKT Systems in Operative Planning for Gas Distribution Networks

AbstractOperative planning in gas networks with prescribed binary decisions yields large scale nonlinear programs defined on graphs. We study the structure of the KKT systems arising in interior methods and present a customized direct solution algorithm. Computational results indicate that the algorithm is suitable for optimization in small and medium‐sized gas networks. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Complex-grid spectral algorithms for inviscid linear instability of boundary-layer flows

We present a suite of algorithms designed to obtain accurate numerical solutions of the generalised eigenvalue problem governing inviscid linear instability of boundary-layer type of flow in both the incompressible and compressible regimes on planar and axisymmetric curved geometries. The large gradient problems which occur in the governing equations at critical layers are treated by diverting the integration path into the complex plane, making use of complex mappings. The need for expansion of the basic flow profiles in truncated Taylor series is circumvented by solving the boundary-layer equations directly on the same (complex) grid used for the instability calculations. Iterative and direct solution algorithms are employed and the performance of the resulting algorithms using nonlinear radiation or homogeneous Dirichlet far-field boundary conditions is examined. The dependence of the solution on the parameters of the complex mappings is discussed. Results of incompressible and supersonic flow examples are presented; their excellent agreement with established works demonstrates the accuracy and robustness of the new methods presented. Means of improving the efficiency of the proposed spectral algorithms are suggested.

Finite element–boundary integral methods in electromagnetics

Mixed sparse–dense, symmetrical–unsymmetrical direct equation solution algorithms are developed, and utilized to fully exploit the special characteristics of the coefficient matrix of system of linear (complex numbers) simultaneous equations. Numerical performance of the developed software is conducted, by performing a simplified Northrop–Grumman finite element–integral equation cavity model on the IBM-R6000/590 workstation.

Efficient Spectral-Galerkin Algorithms for Direct Solution of Second-Order Equations Using Ultraspherical Polynomials

It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of O(N4) (N is the number of retained modes of polynomial approximations). This paper presents some efficient spectral algorithms, which have a condition number of O(N2), based on the ultraspherical-Galerkin methods for second-order elliptic equations in one and two space variables. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. The complexities of the algorithms are a small multiple of Nd+1 operations for a d-dimensional domain with (N-1)d unknowns, while the convergence rates of the algorithms are exponential for problems with smooth solutions.

An efficient parallel procedure for the simulation of crack growth using the cohesive surface methodology

AbstractSimulations of crack growth that are based on the cohesive surface methodology typically involve ill‐conditioned systems of equations and require much processing time. This paper shows how these systems of equations can be solved efficiently by adopting the domain decomposition approach in which the finite element mesh is partitioned into multiple blocks. The system of equations is then reduced to a much smaller system of equations that is solved with an iterative algorithm in combination with a powerful two‐level preconditioner. Although the solution algorithm is more efficient than a direct solution algorithm on a single‐processor computer, it becomes really attractive when used on a parallel computer. This is demonstrated for a large scale simulation of crack growth in a polymer using a Cray T3E with 64 processors. Copyright © 2001 John Wiley & Sons, Ltd.