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Linear approximation
About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.
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Dissertation•
01 May 2013
TL;DR: This thesis considers topology optimization for structural mechanics problems, where the underlying PDE is derived from linear elasticity, and considers the formulation of the SIMP method as a mathematical optimization problem.
Abstract: Topology optimization is a tool for finding a domain in which material is placed that optimizes a certain objective function subject to constraints. This thesis considers topology optimization for structural mechanics problems, where the underlying PDE is derived from linear elasticity. There are two main approaches for solving topology optimization: Solid Isotropic Material with Penalisation (SIMP) and Evolutionary Structural Optimization (ESO). SIMP is a continuous relaxation of the problem solved using a mathematical programming technique and so inherits the convergence properties of the optimization method. By contrast, ESO is based on engineering heuristics and has no proof of optimality. This thesis considers the formulation of the SIMP method as a mathematical optimization problem. Including the linear elasticity state equations is considered and found to be substantially less reliable and less efficient than excluding them from the formulation and solving the state equations separately. The convergence of the SIMP method under a regularising filter is investigated and shown to impede convergence. A robust criterion to stop filtering is proposed and demonstrated to work well in high-resolution problems (O(10^6)). The ESO method is investigated to fully explain its non-monotonic convergence behaviour. Through a series of analytic examples, the steps taken by the ESO algorithm are shown to differ arbitrarily from a linear approximation. It is this difference between the linear approximation and the actual value taken which causes ESO to occasionally take non-descent steps. A mesh refinement technique has been introduced with the sole intention of reducing the ESO step size and thereby ensuring descent of the algorithm. This is shown to work on numerous examples. Extending the classical topology optimization problem to included a global buckling constraint is considered. This poses multiple computational challenges, including the introduction of numerically driven spurious localised buckling modes and ill-defined gradients in the case of non-simple eigenvalues. To counter such issues that arise in a continuous relaxation approach, a method for solving the problem that enforces the binary constraints is proposed. The method is designed specifically to reduce the number of derivative calculations made, which is by far the most computationally expensive step in optimization involving buckling. This method is tested on multiple problems and shown to work on problems of size O(10^5).
21 citations
TL;DR: In this article, an equiripple error constraint was used to design linear approximations to √ x 2 + y 2, where the peak error was less than one percent, which is significantly less than the error obtained by using other design criteria.
Abstract: An equiripple error constraint was used to design linear approximations to \sqrt{x^{2} + y^{2}} . Several examples are presented including one having a peak error of less than one percent, which is significantly less than the error obtained by using other design criteria. The mean and standard deviation of the relative error are also tabulated as are earlier results obtained by other authors.
21 citations
TL;DR: The corner function, a special case of n dimensional sigmoid function, is found to have desirable characteristics, and can be used to approximate functions with much weaker requirements (only boundedness and piecewise continuity).
21 citations
TL;DR: In this paper, a linear equivalent approximation based upon the secant modulus is introduced, from which these experimental data may be predicted, and the author's earlier determination of times of contact and coefficients of restitution from the characteristic of half the maximum stress is shown to be consistent with this linear approximation.
Abstract: A further experimental study is presented of the unloading phenomenon for the constant velocity axial impact of identical annealed aluminium cylindrical rods. Measurements are made of stress-time behaviour at the impact face, displacement-time behaviour at several distances from the impact face as well as at the free end, and strain-time measurements in the vicinity of the free end. A linear equivalent approximation based upon the secant modulus is introduced, from which these experimental data may be predicted. The author's earlier determination of times of contact and coefficients of restitution from the characteristic of half the maximum stress is shown to be consistent with this linear approximation.
21 citations
TL;DR: Improved error bounds are improved on the error bounds derived in Barrett and Blowley (1998) and Blowey (1999c) for a fully practical piecewise linear finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix.
Abstract: Using a slightly different discretization scheme in time and adapting the approach in Nochetto et al. (1998) for analysing the time discretization error in the backward Euler method, we improve on the error bounds derived in (i) Barrett and Blowley (1998) and (ii) Barrett and Blowey (1999c) for a fully practical piecewise linear finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix and (i) a logarithmic free energy, and (ii) a non-smooth free energy (the deep quench limit); respectively. Moreover, the improved error bound in the deep quench limit is optimal. Numerical experiments with three components illustrating the above error bounds are also presented.
21 citations