Application of mathematica to the direct semi-numerical solution of finite element problems

Several numerical methods, such as the finite element method, reduce applied mechanics and additional engineering problems to systems of linear algebraic equations. It has been already suggested that the inclusion of a symbolic parameter in the corresponding numerical results leads to a generality and a wide applicability of these results. Here we suggest the direct solution of these equations by using the popular computer algebra system MATHEMATICA. Assuming the results expressed in a Taylor-Maclaurin series form with respect to the selected symbolic parameter, the whole problem is reduced to the solution of an appropriate number of systems of purely numerical linear equations. This can be achieved either inside MATHEMATICA or by using efficient external numerical routines. As an application the above modification of the finite element method was used in the classical problem of a tapered elastic beam. The obtained semi-numerical results by the finite element method were seen to be in agreement with the available theoretical results. Further possibilities are also suggested in brief.     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

Symbolic finite element analysis using computer algebra: Heat transfer in rectangular duct flow

The problem of heat transfer in fully developed laminar flow in a rectangular duct is solved using a symbolic finite element method. The Nusselt number is obtained as a power series of the aspect ratio of the duct. The solution procedure here differs from the conventional finite element method, in that the aspect ratio remains in symbolic form throughout the computation. Part of the computation is done using the computer algebra system Mathematica. However, the most computational intensive part which involves a Gauss elimination in symbolic form is implemented using an ordinary computer program without resorting to a computer algebra system. The agreement between the results from the present work and those from exact numerical procedures is reasonable.     

Numerical solution of a third-order nonlinear boundary-value problem by automatic differentiation

We develop a simple numerical method for obtaining Taylor series approximation to the solution of a nonlinear third-order boundary-value problem. We use recursive formulas derived from the governing differential equation itself to calculate exact values of the derivatives needed in the Taylor series. Since we do not use difference formulas or symbolic manipulation for calculating the derivatives, our method requires much less computational effort when compared with the techniques previously reported in the literature. We will illustrate the effectiveness of our method with several test problems.     

Computer aided symbolic solution of multi-point boundary value problems occurring in physics and engineering

The author has elsewhere published papers demonstrating applicability of computer algebraic and symbol manipulation in obtaining solutions to ordinary and partial differential equations by Piccard's method, steepest descent and various forms of the Newton-Kantorovich theorems and applying them to non-trivial problems in engineering, physics and, especially, celestial mechanics. In this paper, the Taylor series will be developed permitting expansion about any point and for any boundary conditions for any order derivative at arbitrary points, i.e., the general multi-point boundary valve problem (MPBVP) will be solved. The symbolic algorithm developed is written in PL/1-Formac and produces the Taylor series solution for any nonlinear differential equation in which the highest order derivative may be algebraically isolated. This program permits the continuation of this solution on intervals of the independent variable, in the manner of polynomial splines. This program permits symbolic solutions, e.g. in terms of a symbolic initial condition. However, such a solution requires enormous main storage. PL/1-Formac was used due to its general availability and its compatibility with relatively small main storage, even as small as 200 K bytes. Two parameters are available for attaining a given numerical accuracy; the order of the Taylor expansion and the number of continuation intervals into which the solution range is divided. Experiments show that high accuracy can be obtained by judiciously selecting these two parameters in order to counterbalance truncation error against numerical round-off error. Extensive additional documentation of this procedure has been performed on scores of problems occurring in the applied mathematics literature.     

Logic-based solution methods for optimal control of hybrid systems

Combinatorial optimization over continuous and integer variables is a useful tool for solving complex optimal control problems of hybrid dynamical systems formulated in discrete-time. Current approaches are based on mixed-integer linear (or quadratic) programming (MIP), which provides the solution after solving a sequence of relaxed linear (or quadratic) programs. MIP formulations require the translation of the discrete/logic part of the hybrid problem into mixed-integer inequalities. Although this operation can be done automatically, most of the original symbolic structure of the problem (e.g., transition functions of finite state machines, logic constraints, symbolic variables, etc.) is lost during the conversion, with a consequent loss of computational performance. In this paper, we attempt to overcome such a difficulty by combining numerical techniques for solving convex programming problems with symbolic techniques for solving constraint satisfaction problems (CSP). The resulting "hybrid" solver proposed here takes advantage of CSP solvers for dealing with satisfiability of logic constraints very efficiently. We propose a suitable model of the hybrid dynamics and a class of optimal control problems that embrace both symbolic and continuous variables/functions, and that are tailored to the use of the new hybrid solver. The superiority in terms of computational performance with respect to commercial MIP solvers is shown on a centralized supply chain management problem with uncertain forecast demand.     

用距离几何求解一类几何约束问题

该文设计并实现了一个基于距离几何理论的、系统地求解一类几何约束问题的高效方法,它能够最终给出几何元素的直角坐标表示．应用此方法,给出了八面体问题的符号解以及八面体问题和5P1L问题的一个数值结果．结果显示,该方法与传统的方法相比,具有需要求解方程组的方程个数少、计算效率高、计算彻底、计算机实现方便等优点．     

The decomposition method for Cauchy advection-diffusion problems

In this paper, the solution of Cauchy problems for the advection-diffusion equation is obtained using the decomposition method. In the case when the flow velocity is constant, an analytical solution can be derived, whilst for variable flow velocity, symbolic numerical computations need to be performed.     

A Symbolic-Numerical Method for Finding a Real Solution of an Arbitrary System of Nonlinear Algebraic Equations

In this paper, the author presents a new method for iteratively finding a real solution of an arbitrary system of nonlinear algebraic equations, where the system can be overdetermined or underdetermined and its Jacobian matrix can be of any (positive) rank. When the number of equations is equal to the number of variables and the Jacobian matrix of the system is nonsingular, the method is similar to the well-known Newton's method.The method is a hybrid symbolic-numerical method, in that we utilize some extended procedures from classical computer algebra together with ideas and algorithmic techniques from numerical computation, namely Newton's method and pseudoinverse matrices. First the symbolic techniques are used to transform an arbitrary system of algebraic equations into a set of regular systems. By regular system we mean a system whose Jacobian matrix is of full row rank. Newton-like numerical techniques are then used to find a real solution for each regular system obtained from the symbolic part of the method.The method has a wide range of applicability. It is especially useful for applications in which we need to find some particular solutions from a nonzero-dimensional manifold of real solutions of a system of equations, i.e. the system has infinitely many solutions.We find some mild conditions for the asymptotic convergence of the numerical part of our method. We prove that the asymptotic convergence of the new method is still quadratic while the robustness of the numerical part can be enhanced by techniques like damping as in the regular case. The method has been implemented in Maple andMathematica . Several examples are presented to show that the method works nicely.     

Application of Computer Algebra Systems for Stability Analysis of Difference Schemes on Curvilinear Grids

The paper deals with problems arising in the application of the computer algebra systems for the symbolic–numeric stability analysis of difference schemes and schemes of the finite-volume method approximating the two-dimensional Euler equations for compressible fluid flows on curvilinear spatial grids. We carry out a detailed comparison of the REDUCE 3.6 and Mathematica(Versions 2.2 and 3.0) from the point of view of their applicability to the solution of the above problems. We draw a conclusion that a preference should be given for Mathematica from the viewpoint of the execution of symbolic–numeric computations. We also describe in detail our new symbolic–numeric algorithm for stability investigation, which was implemented with the aid of Mathematica. The proposed method enables us to reduce the needed computer storage at the symbolic stages by a factor of about 20 as compared with the previous algorithms. A feature of the numerical stages is the use of the arithmetic of rational numbers, which enables us to avoid the accumulation of the roundoff errors. We present the examples of the application of the proposed symbolic–numeric method for stability analysis of very complex schemes of the finite-volume method on curvilinear grids, which are widely used in computational fluid dynamics.     

Optimal control of heat conductivity

Consideration was given to the problem of optimal control of heat conductivity in the spherical coordinate system under a bounded control action. Proposed was a procedure of numerical solution of this problem based on the Pontryagin method for the distributed-parameter systems, special iterative process, and new difference schemes for the problem of heat conductivity in the spherical coordinate system. Estimates of the iterative process convergence were proved. The proposed difference schemes and the algorithm on the whole are illustrated by numerical calculations of the test problems.     

Geometric constraint satisfaction using optimization methods

The numerical approach to solving geometric constraint problems is indispensable for building a practical CAD system. The most commonly-used numerical method is the Newton–Raphson method. It is fast, but has the instability problem: the method requires good initial values. To overcome this problem, recently the homotopy method has been proposed and experimented with. According to the report, the homotopy method generally works much better in terms of stability. In this paper we use the numerical optimization method to deal with the geometric constraint solving problem. The experimental results based on our implementation of the method show that this method is also much less sensitive to the initial value. Further, a distinctive advantage of the method is that under- and over-constrained problems can be handled naturally and efficiently. We also give many instructive examples to illustrate the above advantages.     

A testbench of arbitrary accuracy for electromagnetic simulations

Several electromagnetic problems for verification purposes in computational electromagnetics are introduced. Details about the formulation of a generalized eigenvalue problem for non‐lossy and lossy materials are provided to obtain a fast and ready‐to‐use way of verification. Codes written using the symbolic toolbox from MATLAB are detailed to obtain an arbitrary accuracy for the proposed problems. Finally, numerical results in a finite element method code are presented together with the analytical values to show the accuracy of the code proposed.     

A numerical technique for solving fractional optimal control problems

This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs). The fractional derivative in these problems is in the Caputo sense. The method is based upon the Legendre orthonormal polynomial basis. The operational matrices of fractional Riemann-Liouville integration and multiplication, along with the Lagrange multiplier method for the constrained extremum are considered. By this method, the given optimization problem reduces to the problem of solving a system of algebraic equations. By solving this system, we achieve the solution of the FOCP. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

A solution of one-dimensional moving boundaries problems by the finite-element method

A finite-element formulation for the solution of one-dimensional problems with moving boundaries is considered. The movement of the boundaries may not be known a priori. The term “conformal mesh” is defined, and a mesh undergoing conformal deformation is shown to have some numerical advantages. The formulation is applied to linear shape functions. An auxiliary method is suggested, which replaces the repetitive solution of a system of algebraic equations with a one-time solution of an eigenvalue problem. The examples used as models for this method are one- and two-phase Stefan problems and the problem of thermal displacement and stresses in a wall undergoing ablation.     

Computing dynamic output feedback laws

The pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed-loop system has a desired set of eigenvalues. Converting this problem into a question of enumerative geometry, efficient numerical homotopy algorithms to solve this problem for general multiple-input-multiple-output systems have been proposed recently. Despite the wider application range of dynamic feedback laws, the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before. In this note, we present symbolic-numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine. We report on numerical experiments with our publicly available software PHCpack and illustrate its application on various control problems from the literature.     

The problems of accuracy and robustness in geometric computation

Practical implementation of geometric operations remains error-prone, and the goal of implementing correct and robust systems for carrying out geometric computation remains elusive. The problem is variously characterized as a matter of achieving sufficient numerical precision, as a fundamental difficulty in dealing with interacting numeric and symbolic data, or as a problem of avoiding degenerate positions. The author examines these problems, surveys some of the approaches proposed, and assesses their potential for devising complete and efficient solutions. He restricts the analysis to objects with linear elements, since substantial problems already arise in this case. Three perturbation-free methods are considered: floating-point computation, limited-precision rational arithmetic, and purely symbolic representations. Some perturbation approaches are also examined, namely, representation and model, altering the symbolic data, and avoiding degeneracies     

Iterative functional modification method for solving a transportation problem

We propose a new method for solving transportation problems based on decomposing the original problem into a number of two-dimensional optimization problems. Since the solution procedure is integer-valued and monotonic in the objective function, the required computation is finite. As a result, we get not only a single optimal solution of the original transportation problem but a system of constraints that can yield all optimal solutions. We give numerical examples that illustrate the constructions of our algorithm.     

Discovering inequality conditions in the analytic solution of optimization problems

Necessary and sufficient conditions for the problem of maximizing or minimizing a function subject to inequality constraints are given by a set of equalities and inequalities known as the Kuhn-Tucker conditions. These conditions can provide an analytic solution to the optimization problem if the artificial variables known as Lagrange multipliers can be eliminated. However, this is tedious to do by hand. This paper develops a computer program to assist in the solution process which combines symbolic computation and automated reasoning techniques. The program may also be useful for other problems involving algebraic reasoning with inequalities which employ general functions or symbolic parameters.     

Sub-symbolically managing pieces of symbolical functions forsorting

We present a hybrid system for managing both symbolic and sub-symbolic knowledge in a uniform way. Our aim is to solve problems where some gap in formal theories occurs which stops one from getting a fully symbolical solution. The idea is to use neural modules to functionally connect pieces of symbolic knowledge, such as mathematical formulas and deductive rules. The whole system is trained through a backpropagation learning algorithm where all (symbolic or sub-symbolic) free parameters are updated piping back the error through each component of the system. The structure of this system is very general, possibly varying over time and managing fuzzy variables and decision trees. We use as a test-bed the problem of sorting a file, where suitable suggestions on next sorting moves are supplied by the network also on the basis of the hints provided by some conventional sorters. A comprehensive discussion of system performance is provided in order to understand behaviors and capabilities of the proposed hybrid system.