In this paper a block method is developed to use on uniform-graded mesh for the solution of Volterra integral equations of the second kind. This method permits the use of a variable step size when solving Volterra integral equations. Means of reducing the error. Extensive results are presented.

An efficient, accurate, and simple numerical method is necessary for analysis and design of an incompressible potential flow around multi-element airfoils. In this paper, which is the subject of the second part of the study, the mathematical model is built utilizing the local coordinate system, while in the first part of the study [1]. only the global coordinate system is used. Mathematical model, by the vortex panel method with the use of the stream function, is written for the analysis of potential flow over multi-element airfoils. The computational model is built for both uniform and linear vortex distributions with utilizing the constant stream function boundary condition. From the fact that any study which does not consider deeply and precisely, all the parameters relevant to the computational model, might make it fragmentary. Hence, the following parameters are tested to investigate their effect on the accuracy of the method. They are: both types of the vortex distribution, two types of panelling, different ways of applying the Kutta condition, and two ways of positioning the control points. For the purpose of easier comparison, the study cases performed using the present model are restricted only to single-element airfoils; NACA 0012 airfoil at an angle of attack alpha=8.3°, and a cusped trailing edge 15 percent thick Van de Vooren airfoil at alpha=5°. [1] M.M. Bahbah, B. Maruszewski. Effect of panelling and application of the numerical solution of panel methods. ECCS-1 Conference, Changsha, China, 1995.

In this paper a practical procedure for the solution of really sized mixed problems, generating a continuous stress field (where appropriate) and having both the stress and displacement boundary conditions exactly satisfied, is described. The system matrix for the present formulation can be subdivided into the blocks, if the field variables (stresses and displacements) are separated for computational purposes. In addition, the structure of these blocks is sparse, similarly as the structure of the stiffness matrix in classical finite element analysis. Block sparse solution procedure, accounting for the pattern of the resulting system matrix is proposed. Computer implementation confirmed feasibility of the described solution procedure. In addition, numerical tests show remarkably high accuracy and convergence rate of the present mixed scheme for both the stresses and displacements. Due to high accuracy of the scheme, it can be competitive in comparison with usual displacement approach, although the count of arithmetic operations for the same mesh density in mixed procedure can be an order of magnitude larger than in classical finite element analysis.

We investigate the performance of the Naghdi shell model using a family of hierarchic high order finite elements. We solve two cylindrical shell problems, representative of extremely discriminating situations: the membrane dominated Scordelis-Lo problem and a bending dominated problem already tested by Leino and Pitkäranta. As it is well known, these problems are hard tests for shell elements, especially when the thickness of the shell is approaching to zero, since the presence of hidden constraints can lead to numerical convergence problems, known as shear and membrane locking. The numerical results show the robustness of the finite elements developed, able to avoid the locking behavior.

The optimization of the nozzle shape was carried out using the finite element incompressible viscous flow solver, with discretization of total derivative, with the originally developed software. Optimization procedure used conjugate gradient method, with finite difference approximation of gradient of objective function. The mesh generator, specially adapted for chosen shape parametrization in the form of splines using Bezier cubic curve segments, has been used in optimal shape design of the nozzle. The examples of optimization with constraints, the nozzle shape optimization, and the unconstrained optimization of the confusor are presented. All test cases showed good convergence properties that qualifies the proposed methodology as appropriate for shape optimization in viscous flow problems.

In this paper we discuss the use of the singularity subtraction technique incorporated with the Tau Method for the numerical solution of singular partial differential equations which are relevant to the linear elastic fracture mechanics. To treat the singularity, we apply the singularity subtraction technique to the singular boundary value problems. The problems arising in this application are not in the standard form required by the Tau software. By introducing the pseudo-differential equations, lambda'_k = 0, k=1(1)m, to determine the stress intensity and higher order factors lambda_k results in the standard boundary value problems. We consider two model crack problems including Motz' anti-plane crack problem and a plane strain problem defined by the biharmonic equation. We obtain results of considerable accuracy which compare favorably with those published in the recent literature.

The paper presents some aspects of the formulation and numerical implementation of combined mathematical model "elastic body - Timoshenko plate". The variational problem is formulated. The existence of solution of combined model is considered. The numerical investigation of the problem is performed by coupling Direct Boundary Element and Finite Element Methods. Numerical example is presented supporting the analysis.

The paper presents some aspects of the sensitivity analysis within the multivariate distribution models. The presented procedures are provided for engineering problems based on the Nataf model. The Nataf model involves the marginal distributions of the random variables and the correlation between them. Sensitivities are considered through derivatives with respect to the correlation coefficients. The terms for the derivatives of the Nataf correlation coefficients with respect to the given correlation coefficients are presented. The derivatives of the transformations between the random variables are given next. The Cholesky decomposition and the spectral decomposition are applied. Derivatives of the Cholesky decomposition are obtained in the form of a recursive scheme. Derivatives of the eigenvalues and eigenvectors are obtained using perturbations. In addition, a comprehensive method for derivatives of distances and derivatives of angles between the directions is given. Finally, numerical examples are attached to illustrate the presented procedures.

Many numerical methods for studying chemical reaction problems require the computation of the eigenvalues of very large complex symmetric matrices. Recently, a new algorithm for this problem has been proposed by Bar-On and Ryaboy [1]. This algorithm is similar in concept and complexity to the Hermitian eigensolver and is based on application of complex orthogonal transformations to preserve symmetry and recovery transformations to preserve stability. We demonstrate the performance of the proposed algorithm on several high performance computers from Digital, SGI, and Cray. The results show that the new algorithm is much faster than the general eigensolver, the present method used for solving these problems. [1] I. Bar-On, V. Ryaboy. Fast diagonalization of large and dense complex symmetric matrices, with applications to quantum reaction dynamics. SIAM J. on Scientific Computing, 18: 1412--1435, 1997.

We discuss a global, iteration-free numerical scheme (based on the Piecewise Linear algorithm), with special respect to the computation of elasto-plastic frames. The plastic deformations are concentrated in plastic hinges which may appear at both ends of the bars, while the inner parts of the bars can have only elastic deformations but without limation on their magnitude. Our method is tested on a classical example and the results show very good match with those known from the literature. We discuss advantages and disadvantages and point out other, related applications.

New developments in structural analysis methods such as automatic error estimation, adaptive discretization control, and mesh generation are slowly becoming available in engineering practice. On the other hand, graphical interactive user environments and easy-to-use general purpose programs are prolific. Each of these new developments helps to ease the everyday work of the engineer, but only by joining them together in a unified framework and user environment, an entirely new class and generation of structural analysis tools on the computer can be generated. The present study suggests guidelines and principles how such a new generation of software can be brought about, and reports experiences with a prototype application that is already in practical use.