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Register a new userMulti-material phase field approach to structural topology optimization
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Multi-material structural topology and shape optimization problems are formulated within a phase field approach. First-order conditions are stated and the relation of the necessary conditions to classical shape derivatives are discussed. An efficient numerical method based on an $H^1$-gradient projection method is introduced and finally several numerical results demonstrate the applicability of the approach.
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We formulate a general shape and topology optimization problem in structural optimization by using a phase field approach. This problem is considered in view of well-posedness and we derive optimality conditions. We relate the diffuse interface problem to a perimeter penalized sharp interface shape optimization problem in the sense of $Gamma$-convergence of the reduced objective functional. Additionally, convergence of the equations of the first variation can be shown. The limit equations can also be derived directly from the problem in the sharp interface setting. Numerical computations demonstrate that the approach can be applied for complex structural optimization problems.
For the minimization of a nonlinear cost functional $j$ under convex constraints the relaxed projected gradient process $varphi_{k+1} = varphi_{k} + alpha_k(P_H(varphi_{k}-lambda_k abla_H j(varphi_{k}))-varphi_{k})$ is a well known method. The analysis is classically performed in a Hilbert space $H$. We generalize this method to functionals $j$ which are differentiable in a Banach space. Thus it is possible to perform e.g. an $L^2$ gradient method if $j$ is only differentiable in $L^infty$. We show global convergence using Armijo backtracking in $alpha_k$ and allow the inner product and the scaling $lambda_k$ to change in every iteration. As application we present a structural topology optimization problem based on a phase field model, where the reduced cost functional $j$ is differentiable in $H^1cap L^infty$. The presented numerical results using the $H^1$ inner product and a pointwise chosen metric including second order information show the expected mesh independency in the iteration numbers. The latter yields an additional, drastic decrease in iteration numbers as well as in computation time. Moreover we present numerical results using a BFGS update of the $H^1$ inner product for further optimization problems based on phase field models.
This paper introduces progressive algorithms for the topological analysis of scalar data. Our approach is based on a hierarchical representation of the input data and the fast identification of topologically invariant vertices, which are vertices that have no impact on the topological description of the data and for which we show that no computation is required as they are introduced in the hierarchy. This enables the definition of efficient coarse-to-fine topological algorithms, which leverage fast update mechanisms for ordinary vertices and avoid computation for the topologically invariant ones. We demonstrate our approach with two examples of topological algorithms (critical point extraction and persistence diagram computation), which generate interpretable outputs upon interruption requests and which progressively refine them otherwise. Experiments on real-life datasets illustrate that our progressive strategy, in addition to the continuous visual feedback it provides, even improves run time performance with regard to non-progressive algorithms and we describe further accelerations with shared-memory parallelism. We illustrate the utility of our approach in batch-mode and interactive setups, where it respectively enables the control of the execution time of complete topological pipelines as well as previews of the topological features found in a dataset, with progressive updates delivered within interactive times.
Topology optimization has emerged as a popular approach to refine a components design and increasing its performance. However, current state-of-the-art topology optimization frameworks are compute-intensive, mainly due to multiple finite element analysis iterations required to evaluate the components performance during the optimization process. Recently, machine learning-based topology optimization methods have been explored by researchers to alleviate this issue. However, previous approaches have mainly been demonstrated on simple two-dimensional applications with low-resolution geometry. Further, current approaches are based on a single machine learning model for end-to-end prediction, which requires a large dataset for training. These challenges make it non-trivial to extend the current approaches to higher resolutions. In this paper, we explore a deep learning-based framework for performing topology optimization for three-dimensional geometries with a reasonably fine (high) resolution. We are able to achieve this by training multiple networks, each trying to learn a different aspect of the overall topology optimization methodology. We demonstrate the application of our framework on both 2D and 3D geometries. The results show that our approach predicts the final optimized design better than current ML-based topology optimization methods.
This paper provides an extended level set (X-LS) based topology optimiza- tion method for multi material design. In the proposed method, each zero level set of a level set function {phi}ij represents the boundary between materials i and j. Each increase or decrease of {phi}ij corresponds to a material change between the two materials. This approach reduces the dependence of the initial configuration in the optimization calculation and simplifies the sensitivity analysis. First, the topology optimization problem is formulated in the X-LS representation. Next, the reaction-diffusion equation that updates the level set function is introduced, and an optimization algorithm that solves the equilibrium equations and the reaction-diffusion equation using the fi- nite element method is constructed. Finally, the validity and utility of the proposed topology optimization method are confirmed using two- and three- dimensional numerical examples.
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