Numerical solution of creep problems

Numerical techniques which have been previously used in the solution of creep problems are reviewed. The different techniques are used to obtain solutions for the creep deformation and rupture of a steadily loaded three-bar structure. The results of numerical calculations are presented, and the most appropriate techniques are identified for the solution of creep problems in engineering structures.     

Variable mesh methods for the numerical solution of two-point singular perturbation problems

In this paper, variable mesh difference methods of third order are derived for the solution of the two-point, second-order, singular perturbation problems y″ = f(x, y, y′, ε). These methods are applied to a number of examples and their superiority over the existing methods is shown.     

A unified framework for the numerical solution of optimal control problems using pseudospectral methods

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Transformations are developed that relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. The LG and LGR discrete costate systems are full rank while the LGL discrete costate system is rank-deficient. The LGL costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the LGL discrete costate system. An example is considered to assess the accuracy and features of each collocation scheme.     

Unified framework for numerical methods to solve the time-dependent Maxwell equations

We present a comparative study of numerical algorithms to solve the time-dependent Maxwell equations for systems with spatially varying permittivity and permeability. We show that the Lie-Trotter-Suzuki product-formula approach can be used to construct a family of unconditionally stable algorithms, the conventional Yee algorithm, and two new variants of the Yee algorithm that do not require the use of the staggered-in-time grid. We also consider a one-step algorithm, based on the Chebyshev polynomial expansion, and compare the computational efficiency of the one-step, the Yee-type, the alternating-direction-implicit, and the unconditionally stable algorithms. For applications where the long-time behavior is of main interest, we find that the one-step algorithm may be orders of magnitude more efficient than present multiple time-step, finite-difference time-domain algorithms.     

High order A-stable numerical methods for stiff problems

Stiff problems pose special computational difficulties because explicit methods cannot solve these problems without severe limitations on the stepsize. This idea is illustrated using a contrived linear test problem and a discretized diffusion problem. Even though the Euler method can solve these problems if the stepsize is small enough, there is no such limitation for the implicit Euler method. To obtain high order A-stable methods, it is traditional to turn to Runge-Kutta methods or to linear multistep methods. Each of these has limitations of one sort or another and we consider, as a middle ground, the use of general linear (or multivalue multistage) methods. Methods possessing the property of inherent Runge-Kutta stability are identified as promising methods within this large class, and an example of one of these methods is discussed. The method in question, even though it has four stages, out-performs the implicit Euler method if sufficient accuracy is required, because of its higher order.     

Improved solution methods for inelastic rate problems

The objective of the paper is an assessment of the incremental solution methods for the analysis of inelastic rate problems. In particular, the possibilities of the initial load method are explored to improve the accuracy and stability of the traditional explicit operators by higher-order time expansions and implicit weighting schemes.The convergence limitations are examined for different classes of inelastic growth laws (viscous flow, viscoelasticity, viscoplasticity) which restrict the time step because of the iterative solution of the implicit algorithm. The range and rate of convergence of the initial load method (constant stiffness predictor-corrector iteration) is enlarged by tangential gradient techniques which account for the inelastic response in the structural stiffness matrix. In this way the time step restriction disappears although at a considerable increase of computational expense because of the costly computation and decomposition of structural gradients within each iteration cycle (Newton-Raphson methods).As compared to the linear single-step methods, the cubic Hermitian time expansions furnish far better accuracy than the traditional linear expansions for very little increase of computational cost. Stability and convergence limits correspond to those of the lower-order operators, whereby the implicit midstep of backward weighting schemes are most advantageous. In this context it is worth noting that aging or strain-hardening effects in the inelastic growth law reduce dramatically the time step restrictions of the iterative initial load solution methods (predictor-corrector schemes), as compared to the simplest creep model in which the inelastic growth law depends only on stress, e.g. for viscous flow and viscoplasticity.     

Block iterative methods for the numerical solution of three dimensional mildly non-linear biharmonic problems of first kind

In this article, we discuss two sets of new finite difference methods of order two and four using 19 and 27 grid points, respectively over a cubic domain for solving the three dimensional nonlinear elliptic biharmonic problems of first kind. For both the cases we use block iterative methods and a single computational cell. The numerical solution of (?u/?n) are obtained as by-product of the methods and we do not require fictitious points in order to approximate the boundary conditions. The resulting matrix system is solved by the block iterative method using a tri-diagonal solver. In numerical experiments the proposed methods are compared with the exact solutions both in singular and non-singular cases.     

General formulation for the numerical solution of optimal control problems

A new improved computational method for a class of optimal control problems is presented. The state and the costate (adjoint) variables are approximated using a set of basis functions. A method, similar to a variational virtual work approach with weighing coefficients, is used to transform the canonical equations into a set of algebraic equations. The method allows approximating functions that need not satisfy the initial conditions a priori. A Lagrange multiplier technique is used to enforce the terminal conditions. This enlarges the space from which the approximating functions can be chosen. Orthogonal polynomials are used to obtain a set of simultaneous equations with fewer non-zero entries. Such a sparse system results in substantial computational economy. Two examples, a time-invariant system and a time-varying system with quadratic performance index, are solved using three different sets of orthogonal polynomials and the power series to demonstrate the feasibility and efficiency of this method.     

An improved numerical process for solution of solid mechanics problems

This paper gives an overview of the development and status of an improved numerical process for the solution of solid mechanics problems. The proposed process uses a mixed formulation with the fundamental unknowns consisting of both stress and displacement parameters. The problem is formulated either by means of first-order partial differential equations or in a variational form by using a Hellinger-Reissner-type mixed variational principle.

For presentation purposes, the components of a numerical process are characterized and the criteria for an ideal process are outlined. Commonly used finite-difference and finite-element procedures arc examined in the light of these criteria and it is shown that they fall short in a number of ways. The proposed numerical process, on the other hand, satisfies most of the optimality criteria and appears to be particularly suited for use with the forthcoming generation computers (e.g. STAR-100 computer).

The paper includes a number of examples showing application of the proposed process to a broad spectrum of solid mechanics problems. These examples demonstrate the versatility and high accuracy of the numerical process obtained by using mixed formulations in conjunction with improved discretization techniques.     

Evaluation of a class of iterative methods for the numerical solution of Boundary value problems for ordinary differential equations

The aim of this paper is an experimental evaluation and comparison among several iterative methods proposed for approximate solution of two-point boundary value problems for ordinary differential equations.     

A stabilizing transformation for numerical solution of maximum principle problems

To aid in solving numerically a two-point boundary problem, a transformation of variables is proposed. This transformation is useful when an analog computer is being used to solve the differential equations, for it keeps the computations on scale and reduces the sensitivity of terminal conditions to initial conditions. Its performance in the presence of computational error appears satisfactory.     

Hybrid numerical methods for convection-diffusion problems in arbitrary geometries

The hybrid nodal-integral/finite element method (NI-FEM) and the hybrid nodal-integral/finite analytic method (NI-FAM) are developed to solve the steady-state, two-dimensional convection-diffusion equation (CDE). The hybrid NI-FAM for the steady-state problem is then extended to solve the more general time-dependent, two-dimensional, CDE. These hybrid coarse mesh methods, unlike the conventional nodal-integral approach, are applicable in arbitrary geometries and maintain the high efficiency of the conventional nodal-integral method (NIM). In steady-state problems, the computational domain for both hybrid methods is discretized using rectangular nodes in the interior of the domain and along vertical and horizontal boundaries, while triangular nodes are used along the boundaries that are not parallel to the x or y axes. In time-dependent problems, the rectangular and triangular nodes become space-time parallelepiped and wedge-shaped nodes, respectively. The difference schemes for the variables on the interfaces of adjacent rectangular/parallelepiped nodes are developed using the conventional NIM. For the triangular nodes in the hybrid NI-FEM, a trial function is written in terms of the edge-averaged concentration of the three edges and made to satisfy the CDE in an integral sense. In the hybrid NI-FAM, the concentration over the triangular/wedge-shaped nodes is represented using a finite analytic approximation, which is based on the analytic solution of the one-dimensional CDE. The difference schemes for both hybrid methods are then developed for the interfaces between the rectangular/parallelepiped and triangular/wedge-shaped nodes by imposing continuity of the flux across the interfaces. A formal derivation of these hybrid methods and numerical results for several test problems are presented and discussed.     

Two methods for the numerical solution of Bueckner's singular integral equation for plane elasticity crack problems

The classical collocation and Galerkin methods are used for the numerical solution of singular integral equations of the first kind involving a finite-part integral with a double pole singularity. Such equations appear in plane elasticity crack problems, where they were suggested by Bueckner, and the unknown function in them is proportional to the crack opening displacement function. An application of the proposed methods to the problem of a straight crack under an exponential normal loading distribution is also made and shows the rapid convergence of the obtained numerical results for the stress intensity factors at the crack tips to their theoretical values.     

Analytic numerical solution of coupled semi-infinite diffusion problems

In this paper, we consider coupled semi-infinite diffusion problems of the form u_t(x, t)− A² u_xx(x,t) = 0, x> 0, t> 0, subject to u(0,t)=B and u(x,0)=0, where A is a matrix in , and u(x,t), and B are vectors in . Using the Fourier sine transform, an explicit exact solution of the problem is proposed. Given an admissible error and a domain D(x₀,t₀)={(x,t);0≤x≤x₀, t≥t₀ > 0, an analytic approximate solution is constructed so that the error with respect to the exact solution is uniformly upper bounded by in D(x₀, t₀).     

On the optimisation of viscoplastic constitutive modelling using a numerical feedback damping algorithm

The authors propose a mathematical model that, in the presence of a constant time step algorithm and a smooth evolution of a state variable, increases the performance of the numerical process, forces the convergence of the numerical solution and, consequently, improves the overall quality of the results. This method reduces the total number of time steps of a simulation process and minimizes the necessary CPU time. The proposed algorithm is based on the mathematical adjustment of the evolution of a chosen state variable. The resulting numerical signals are analysed and properly characterised. Several numerical signals, obtained from different non-linear simulation examples and conditions, are studied. Based on the characterisation of the numerical signals and considering that the numerical results reflect the behaviour of a vibratory system—the numerical code—with its own intrinsic mass, spring and dashpot elements, the authors develop a numerical damping algorithm and present its implementation. The algorithm is applied and tested with a non-linear finite element example, using a viscoplastic constitutive model. The authors also present a set of numerical validation tests consisting of the simulation of the development of residuals stresses that arise from the fabrication process of particle reinforced metal matrix composites (MMC). The cooling down stage of an AlSiC_p 20% vol. MMC is simulated. In order to evaluate the performance of the algorithms, some results, obtained with and without the application of the optimisation algorithm, are presented and thoroughly compared. The numerical damper algorithm proves to be very efficient.     

Initial value problems: numerical methods and mathematics

A number of questions and results concerning Runge-Kutta and general linear methods are surveyed. These include order conditions and order bounds for Runge-Kutta methods, the A-stability of implicit Runge-Kutta methods based on Gaussian quadrature and transformation methods of implementation which lead to singly-implicit methods. The sections dealing with general linear methods include a discussion of the order conditions and an algebraic structure for carrying out order analyses as well as an introduction to a special function associated with parallel methods for stiff problems.     

On the numerical solution of nonlinear eigenvalue problems

We consider the numerical solution of the nonlinear eigenvalue problemA(λ)x=0, where the matrixA(λ) is dependent on the eigenvalue parameter λ nonlinearly. Some new methods (the BDS methods) are presented, together with the analysis of the condition of the methods. Numerical examples comparing the methods are given.