Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
A simple level set method for solving Stefan problems
Journal of Computational Physics
SIAM Review
A PDE-based fast local level set method
Journal of Computational Physics
Shapes and geometries: analysis, differential calculus, and optimization
Shapes and geometries: analysis, differential calculus, and optimization
Variational problems and partial differential equations on implicit surfaces
Journal of Computational Physics
Some Improvements of the Fast Marching Method
SIAM Journal on Scientific Computing
Etude de Problème d'Optimal Design
Proceedings of the 7th IFIP Conference on Optimization Techniques: Modeling and Optimization in the Service of Man, Part 2
Transport and diffusion of material quantities on propagating interfaces via level set methods
Journal of Computational Physics
An Eulerian Formulation for Solving Partial Differential Equations Along a Moving Interface
Journal of Scientific Computing
Structural optimization using sensitivity analysis and a level-set method
Journal of Computational Physics
Journal of Computational Physics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Velocity Extension for the Level-set Method and Multiple Eigenvalues in Shape Optimization
SIAM Journal on Control and Optimization
Numerical Methods for Differential Equations in Random Domains
SIAM Journal on Scientific Computing
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
Maintaining the point correspondence in the level set framework
Journal of Computational Physics
A fictitious domain approach to the numerical solution of PDEs in stochastic domains
Numerische Mathematik
Shape feature control in structural topology optimization
Computer-Aided Design
Structural and Multidisciplinary Optimization
Robust topology optimization accounting for misplacement of material
Structural and Multidisciplinary Optimization
On robust design optimization of truss structures with bounded uncertainties
Structural and Multidisciplinary Optimization
On the similarities between micro/nano lithography and topology optimization projection methods
Structural and Multidisciplinary Optimization
Level-set methods for structural topology optimization: a review
Structural and Multidisciplinary Optimization
A survey of structural and multidisciplinary continuum topology optimization: post 2000
Structural and Multidisciplinary Optimization
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Geometric uncertainty refers to the deviation of the geometric boundary from its ideal position, which may have a non-trivial impact on design performance. Since geometric uncertainty is embedded in the boundary which is dynamic and changes continuously in the optimization process, topology optimization under geometric uncertainty (TOGU) poses extreme difficulty to the already challenging topology optimization problems. This paper aims to solve this cutting-edge problem by integrating the latest developments in level set methods, design under uncertainty, and a newly developed mathematical framework for solving variational problems and partial differential equations that define mappings between different manifolds. There are several contributions of this work. First, geometric uncertainty is quantitatively modeled by combing level set equation with a random normal boundary velocity field characterized with a reduced set of random variables using the Karhunen---Loeve expansion. Multivariate Gauss quadrature is employed to propagate the geometric uncertainty, which also facilitates shape sensitivity analysis by transforming a TOGU problem into a weighted summation of deterministic topology optimization problems. Second, a PDE-based approach is employed to overcome the deficiency of conventional level set model which cannot explicitly maintain the point correspondences between the current and the perturbed boundaries. With the explicit point correspondences, shape sensitivity defined on different perturbed designs can be mapped back to the current design. The proposed method is demonstrated with a bench mark structural design. Robust designs achieved with the proposed TOGU method are compared with their deterministic counterparts.