Exact and Accurate Solutions in the Approximate Reanalysis of Structures

Abstract

Combined approximations (CA) is an efe cient method for reanalysis of structures where binomial series terms are used as basis vectors in reduced basis approximations. In previous studies high-quality approximations have been achieved for large changes in the design, but the reasons for the high accuracy were not fully understood. In this work some typical cases, where exact and accurate solutions are achieved by the method, are presented and discussed. Exact solutions are obtained when a basis vector is a linear combination of the previous vectors. Such solutions are obtained also for low-rank modie cations to structures or scaling of the initial stiffness matrix. In general the CA method provides approximate solutions, but the results presented explain the high accuracy achieved with only a small number of basis vectors. Accurate solutions are achieved in many cases where the basis vectors come close to being linearly dependent. Such solutions are achieved also for changes in a small number of elements or when the angle between the two vectors representing the initial design and modie ed design is small. Numerical examples of various changes in cross sections of elements and in the layout of the structure show that accurate results are achieved even in cases where the series of basis vectors diverges. I. Introduction M ULTIPLErepeatedanalysesareneeded invariousdesign and optimization problems. In general, the structural response cannot be expressed explicitly in terms of the structure properties, and structural analysis involves the solution of a set ofsimultaneous equations. Reanalysis methods are intended to efe ciently analyze structures modie ed due to changes in the design. Approximate reanalysis methods have been used extensively in structural optimization to reduce the number of exact analyses and the overall computational cost during the solution process. The combined approximations (CA) method developed recently is considered in this paper. The method combines several concepts and methods such as reduced basis, series approximations, matrix factorization and Gram ‐Schmidt orthonormalization. These and other methods are used to achieve effective solution procedures. The effectivenessofthemethodinvariousoptimizationproblemshasbeen demonstrated in previous studies. 1i5 Initially the CA method was used only for linear reanalysis models. Recently, the method has been used successfully also in eigenvalue 6 and nonlinear analysis 7 problems. Applications of the method in a large variety ofproblems are discussed elsewhere. 8i11 High-quality approximations of the structural response for large changes in the design have been achieved in previous studies, but the reasons for the high accuracy were not fully understood. In this paper some typical cases, where exact and accurate solutions are achieved by the CAmethod, are presented anddiscussed.In general the CA method provides exact solutions, but the results presented in the paper explain the high accuracy achieved with only a small number of basis vectors.The solution procedure isbriee y described in Sec. II. Three typical cases, where exact solutions are achieved by the CA method, are introduced and discussed in Sec. III. Exact solutions are obtained when a basis vector is a linear combination of the previous vectors. Such solutions are obtained also for lowrank modie cations to structures or scaling of the initial stiffness matrix. Various cases of accurate solutions are discussed in Sec. IV. Convergence properties of the series of basis vectors and the series of the CA terms are presented, and criteria intended to evaluate