Users:Structural Optimization/Optimization Algorithms

Contents 1 General Remarks to Optimization Algorithms 1.1 Available Unconstrained Optimization Algorithms 1.2 Available Constrained Optimization Algorithms 2 Line Search 3 Convergence Measures 4 Filtering 5 Mesh Regularization

General Remarks to Optimization Algorithms

Optimization algorithms are applied to solve optimization problems by a specific iterative procedure. In general, these algorithms are separated in constrained and unconstrained optimization algorithms. Unconstrained algorithms are applicable to unconstrained optimization problems without constraints. Whenever constraints have to be incorporated, constraint optimization algorithms have to be applied. Usually constrained optimization problems are more difficult to solve than unconstrained problems.

Any optimization algorithm computes for each step k a search direction s^k. The new design x^k+1 follows from x^k+1 = x^k + α'^k * s^k. Here the parameter α'^k specifies the actual step length computed by a line search algorithm.

Carat++ solves optimization problems with FE-based parametrization. This means that the design variables are coupled to finite elements, e.g. element thickness, nodal coordinates, etc. Such optimization problems have a large number of design variables and the necessary derivatives are usually not smooth enough. This problem is solved by filter algorithms.

FE-based structural optimization additionally requires a good mesh quality to ensure precise sensitivity analysis. This is ensured by mesh regularization methods.

Available Unconstrained Optimization Algorithms

• Steepest Descent Method
• Conjugate Gradient Method

Available Constrained Optimization Algorithms

• Method of Feasible Directions
• Augmented Lagrange Multiplier Method

Line Search

Each optimization strategy computes a search direction according to a specified algorithm. The line search specifies a suitable step length parameter that minimizes the objective if the design is changed according to the search direction. In Carat++ several line search methods are implemented. More information can be found on page Users:Structural Optimization/Optimization Algorithms/Line Search.

Convergence Measures

Convergence of optimization problems is checked by a specific module according to several criteria. More information is presented on page Users:Structural Optimization/Optimization Algorithms/Convergence Measures

Filtering

One advantage of finite element based optimization is the huge designspace giving the most possible freedom to the optimization algorithm. Unfortunately this advantage might turn out as a drawback, too. As the design space is very big, the algorithm is enabled to produce short-waved solutions which usually are not intended by the user. For this reason a smoothing has to be applied to the gradient field in order to obtain a minimum wave length inside the design update.

Details concerning filtering can be found on the page Users:Structural Optimization/Optimization Algorithms/Filtering.

Mesh Regularization

FE-based shape optimization requires reliable finite elements with high quality. The element shape is one of the most important quantities that determine the robustness of element responses. Therefore, it is extremely important to run shape optimization problems with simultaneous mesh regularization. Carat++ offers geometrical and mechanical methods for mesh regularization. More information is presented on page Users:General FEM Analysis/Mesh regularization.

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