The paper presents an efficient 88 line MATLAB code for topology optimization. It has been developed using the 99 line code presented by Sigmund (Struct Multidisc Optim 21(2):120---127, 2001) as a starting point. The original code has been extended by a density filter, and a considerable improvement in efficiency has been achieved, mainly by preallocating arrays and vectorizing loops. A speed improvement with a factor of 100 is obtained for a benchmark example with 7,500 elements. Moreover, the length of the code has been reduced to a mere 88 lines. These improvements have been accomplished without sacrificing the readability of the code. The 88 line code can therefore be considered as a valuable successor to the 99 line code, providing a practical instrument that may help to ease the learning curve for those entering the field of topology optimization. The paper also discusses simple extensions of the basic code to include recent PDE-based and black-and-white projection filtering methods. The complete 88 line code is included as an appendix and can be downloaded from the web site www.topopt.dtu.dk .

Efficient topology optimization in MATLAB using 88 lines of code

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Nonlinear Functional Analysis And Its Applications

Rate-independent systems occur as limit problems in many physical and mechanical problems, if the interesting time scales are much longer than the intrinsic time scales in the system. Rate-independent systems are sometimes also called “quasistatic systems,” however, the term “quasistatic” is often used in a more general sense—namely, if the inertial terms in a system are neglected, but viscous effects might still be present. The chapter discusses different aspects of rate-independent models or hysteresis operators in the context of continuum mechanics. There are several areas in these fields that have evolved quite independently and have developed their own languages and notation. The chapter compares these different approaches by translating them into one language, and thus provides a useful overview of the different methods in the field.

Chapter 6 - Evolution of Rate-Independent Systems

A mathematical model for a finite–strain elastoplastic evolution problem is proposed in which one time–step of an implicit time–discretization leads to generally non–convex minimization problems. The elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations. The results for a single slip–system and von Mises plasticity illustrate that finite–strain elastoplasticity generates reduced problems with non–quasiconvex energy densities and so allows for non–attainment of energy minimizers and microstructures.

/pdf/non-convex-potentials-and-microstructures-in-finite-strain-rzwthbepkz.pdf

Non-convex potentials and microstructures in finite-strain plasticity

We propose a rate-independent, mesoscopic model for the hysteretic evolution of phase transformations in shape-memory alloys. The model uses the deformation and phase-indicator function as basic unknowns and the potentials for the elastic energy and for the dissipation as constitutive laws. Using the associated functionals, admissible processes are defined to be the ones which are stable at all times and which satisfy the energy inequality.

/pdf/a-variational-formulation-of-rate-independent-phase-2h0uvgzwkc.pdf

A Variational Formulation of Rate-Independent Phase Transformations Using an Extremum Principle

A short Matlab implementation forP1-Q1 finite elements on triangles and parallelograms is provided for the numerical solution of elliptic problems with mixed boundary conditions on unstructured grids. According to the shortness of the program and the given documenta- tion, any adaptation from simple model examples to more complex problems can easily be performed. Numerical examples prove the flexibility of the Matlab tool. Unlike complex black-box commercial computer codes, this paper provides a simple and short open-box Matlab implementation of combined Courant's P1 triangles and Q1 elements on parallelograms for the numerical solutions of elliptic problems with mixed Dirichlet and Neumann boundary conditions. Based on four or five data files, arbitrary regular triangulations are determined. Instead of covering all kinds of possible problems in one code, the proposed tool aims to be simple, easy to understand and modify. Therefore, only simple model examples are included to be adapted to whatever is needed. In further contributions we provide more complicated elements, a posteriori error estimators and flexible adaptive mesh-refining algorithms. Compared to the latest Matlab toolbox (3), our approach is shorter, allows more elements, is easily adapted to modified problems like convection terms, and is open to easy modifications for basically any type of finite element. The rest of the paper is organised as follows. As a model problem, the Laplace

/pdf/remarks-around-50-lines-of-matlab-short-finite-element-2clmphak7i.pdf

Remarks around 50 lines of Matlab: short finite element implementation

A short Matlab implementation for P1 and Q1 finite elements (FE) is provided for the numerical solution of 2d and 3d problems in linear elasticity with mixed boundary conditions. Any adaptation from the simple model examples provided to more complex problems can easily be performed with the given documentation. Numerical examples with postprocessing and error estimation via an averaged stress field illustrate the new Matlab tool and its flexibility.

/pdf/matlab-implementation-of-the-finite-element-method-in-1tft6k53n6.pdf

Matlab implementation of the finite element method in elasticity

Adaptive numerical analysis in primal elastoplasticity with hardening

The quasi-static elastoplastic evolution problem with combined isotropic and kinematic hardening is considered with emphasis on optimal convergence of the lowest order scheme. In each time-step of a generalized midpoint scheme such as the implicit Euler or the Crank--Nicolson scheme, the spatial discretization consists of minimizing a convex but nonsmooth function on a subspace of continuous piecewise linear, resp., piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which, for smooth data and a smooth exact solution, proves linear convergence as the mesh-size tends to zero. Strong convergence of the time-derivatives is established under mild conditions on the mesh- and time-step sizes. Numerical experiments confirm our theoretical predictions on the improved spatial convergence and indicate that the Crank--Nicolson scheme is not always superior over the implicit Euler scheme in practice.

Numerical Analysis of Time-Depending Primal Elastoplasticity with Hardening

https://www.math.hu-berlin.de/~cc/cc_homepage/download/2003-CC_AJ-Averaging_Techniques_Elastoplasticity_with_Hardening.pdf

Jochen Alberty

Papers

Remarks around 50 lines of Matlab: short finite element implementation

Matlab implementation of the finite element method in elasticity

Adaptive numerical analysis in primal elastoplasticity with hardening

Numerical Analysis of Time-Depending Primal Elastoplasticity with Hardening

Averaging techniques for reliable a posteriori FE-error control in elastoplasticity with hardening