Combined Approximations Approach Research Articles

A Simple Matlab Code for Material Design Optimization Using Reduced Order Models.

The main part of the computational cost required for solving the problem of optimal material design with extreme properties using a topology optimization formulation is devoted to solving the equilibrium system of equations derived through the implementation of the finite element method (FEM). To reduce this computational cost, among other methodologies, various model order reduction (MOR) approaches can be utilized. In this work, a simple Matlab code for solving the topology optimization for the design of materials combined with three different model order reduction approaches is presented. The three MOR approaches presented in the code implementation are the proper orthogonal decomposition (POD), the on-the-fly reduced order model construction and the approximate reanalysis (AR) following the combined approximations approach. The complete code, containing all participating functions (including the changes made to the original ones), is provided.

Efficient non‐linear reanalysis of skeletal structures using combined approximations

AbstractNon‐linear reanalysis of large‐scale structures usually involves much computational effort, because the set of non‐linear equations must be solved repeatedly during the solution process. Various approximations that are often used for linear reanalysis are not sufficiently accurate for non‐linear problems. In this study, solution procedures based on the combined approximations approach are developed and compared in terms of efficiency and accuracy. Various path‐independent non‐linear analysis and reanalysis problems are considered, including material non‐linearity, geometric non‐linearity and buckling analysis. Numerical examples demonstrate the effectiveness of the procedures presented. It is shown that in various cases accurate results can be achieved efficiently. Copyright © 2007 John Wiley & Sons, Ltd.

Efficient procedures for repeated calculations of structural sensitivities

Procedures for sensitivity analysis of large-scale structures are presented. Analytical derivatives are evaluated for multiple modified designs where the exact response is unknown. Approximation concepts are used to improve the efficiency and the ease-of-implementation of the calculations. The procedures presented are based on explicit approximations of the response and the combined approximations approach, which is further developed to improve the efficiency of the calculations and the accuracy of the results. Considering sensitivities of static problems and eigenproblems, basis vectors are calculated for a limited number of design points, and then used to evaluate sensitivities for various modified designs. The procedures presented are compared with exact analytical sensitivities. It is shown that accurate results can be achieved with significant reductions in the computational effort.

Efficient Dynamic Reanalysis of Structures

The combined approximations approach, developed originally for linear static reanalysis, is used for dynamic reanalysis of structures. The approach is based on the integration of several concepts and methods, including series expansion, reduced basis, matrix factorization, and Gram-Schmidt orthogonalizations. The advantage is that efficient local approximations and accurate global approximations are combined to achieve an effective solution procedure. Considering modal analysis, the computational effort involved in reanalysis is significantly reduced. However, the accuracy of the higher mode shapes might be insufficient. A procedure intended to improve the accuracy of the results is developed. A reduced eigenproblem is introduced, and a method for determining the basis vectors is presented. Improved basis vectors, using the concepts of shifts and Gram-Schmidt orthogonalizations, are developed, and eigenproblem reanalysis by combined approximations using inverse iteration with shifts is introduced. Numerical examples demonstrate the quality of the results. It is shown that high accuracy is achieved efficiently for various types of changes in the structure.

Nonlinear Dynamic Sensitivities of Structures Using Combined Approximations

Calculation of design sensitivities involves much computational effort, particularly for nonlinear dynamic response of large structures. Modal analysis is often not effective for such problems, because eigenproblems must be solved many times. Approximation concepts, which are used to reduce the computational cost involved in structural analysis, are usually not sufficiently accurate for sensitivity analysis. In this study, approximate reanalysis concepts are used to improve the efficiency of nonlinear dynamic sensitivity analysis. Efficient evaluation of the derivatives, using modal analysis, finite-difference of eigenpairs and the recently developed combined approximations approach, is presented. Several approximate solution procedures are developed and compared in terms of the efficiency and the accuracy of the results. It is shown that some of the approximations presented reduce significantly the computational effort and provide sufficiently accurate results. I. Introduction Design sensitivity analysis of structures deals with the calculation of the response derivatives with respect to the design variables. These derivatives are used in the solution of various problems such as design optimization, generation of response approximations, including approximate reanalysis models, and explicit approximations of the constraint functions in terms of the structural parameters. Sensitivities are required also for assessing the effects of uncertainties in the structural properties on the system response. Calculation of design sensitivities involves much computational effort, particularly for nonlinear dynamic response of large structures. As a result, there has been much interest in efficient procedures for calculating the sensitivity coefficients. Early and recent developments in methods for sensitivity analysis are discussed in various studies. 1-4 Methods of sensitivity analysis for discretized systems can be divided into the following classes: a. Finite-difference methods, which are easy to implement but might involve numerous repeated analyses and high computational cost, particularly in large problems. The efficiency can be improved by using fast reanalysis techniques, but finite-difference approximations might have accuracy problems. b. Analytical methods, which provide exact solutions but might not be easy to implement in some problems. c. Semi-analytical methods, which are based on a compromise between finite-difference methods and analytical methods. These methods are easy to implement but might provide inaccurate results. In general, the main factors considered in choosing a suitable sensitivity analysis method for a specific application include the accuracy, the computational effort, and the ease-of-implementation. The quality of the results and efficiency of the calculations are usually two conflicting factors. That is, higher accuracy is often achieved at the expense of more computational effort. Dynamic sensitivity analysis has been demonstrated by several authors. In Ref. 5 the mode superposition approach has been considered for linear response. Assuming harmonic loading, the response sensitivities were evaluated by direct differentiation of the equations of motion. In several studies

Efficient procedures for repeated calculations of the structural response using combined approximations

Efficient procedures for repeated calculations of the response of large-scale structures that involve numerous repeated analyses are presented. These procedures are based on the combined approximations approach, which is further developed to improve both the efficiency of the calculations and the accuracy of the results. The procedures are suitable for linear, nonlinear, static, and dynamic reanalysis problems, and they can be used to improve the efficiency of response surface calculations. It is shown that the procedures presented provide accurate approximations and may significantly reduce the computational effort involved in reanalysis of large-scale structures. This might prove useful in various design and optimization problems.

Nonlinear dynamic reanalysis of structures by combined approximations

A major difficulty in non-linear dynamic reanalysis by mode superposition is the need to repeat the eigenproblem solutions. This study shows how the combined approximations approach can be used to overcome this difficulty. The approach is based on the integration of several concepts and methods, including matrix factorization, series expansion and reduced basis. The advantage is that efficient local approximations and accurate global approximations are combined to achieve an effective solution procedure. In the procedure developed for material non-linearity, improved basis vectors are introduced using the concepts of Gram–Schmidt orthogonalizations and shifts. Numerical examples show that accurate approximations are achieved efficiently for very large changes in the design variables.

Efficient design sensitivities of structures subjected to dynamic loading

Calculation of design sensitivities often involves much computational effort, particularly in large structural systems with many design variables. Approximation concepts, which are often used to reduce the computational cost involved in repeated analysis, are usually not sufficiently accurate for sensitivity analysis. In this study, approximate reanalysis is used to improve the efficiency of dynamic sensitivity analysis. Using modal analysis, the response derivatives with respect to design variables are presented as a combination of sensitivities of the eigenvectors and the generalized displacements. A procedure intended to reduce the number of differential equations that must be solved during the solution process is proposed. Efficient evaluation of the derivatives, using finite difference and the recently developed combined approximations approach, is presented. Numerical examples show that high accuracy of design sensitivities can be achieved efficiently.

Structural reanalysis for topological modifications - a unified approach

A unified approach for structural reanalysis of all types of topological modifications is presented. The modifications considered include various cases of deletion and addition of members and joints. The most challenging problem where the structural model is itself allowed to vary is presented. The two cases, where the number of degrees of freedom is decreased and increased, are considered. Various types of modified topologies are discussed, including the common conditionally unstable structures. The solution procedure is based on the combined approximations approach and involves small computational effort. Numerical examples show that accurate results are achieved for significant topological modifications. Exact solutions are obtained efficiently for modifications in a small number of members.