SUMMARYA methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described. Nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology. The projection is made via mesh independent functions that are based upon the minimum length scale. A simple linear projection scheme and a non-linear scheme using a regularized Heaviside step function to achieve nearly 0 -1 solutions are examined. The new approach is demonstrated on the minimum compliance problem and the popular SIMP method is used to penalize the stiffness of intermediate volume fraction elements. Solutions are shown to meet user-defined length scale criterion without additional constraints, penalty functions or sensitivity filters. No instances of mesh dependence or checkerboard patterns have been observed.
SUMMARYA new methodology is proposed for the topology optimization of fluid in Stokes flow. The binary design variable and no-slip condition along the solid-fluid interface are regularized to allow for the use of continuous mathematical programming techniques. The regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by Darcy's law. Fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the Stokes and Darcy equations. The mixed formulation of the new Darcy-Stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems. Convergence to the no-slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid-fluid interface are not needed. The optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature. The advantages of the Darcy-Stokes approach include that it uses existing stabilization techniques to solve the finite element problem, it produces 0 -1 (void-solid) topologies (i.e. there are no regions of artificial material), and that it can potentially be used to optimize the layout of a microscopically porous material.
Topology optimization is used to systematically design periodic materials that are optimized for multiple properties and prescribed symmetries. In particular, mechanical stiffness and fluid transport are considered. The base cell of the periodic material serves as the design domain and the goal is to determine the optimal distribution of material phases within this domain. Effective properties of the material are computed from finite element analyses of the base cell using numerical homogenization techniques. The elasticity and fluid flow inverse homogenization design problems are formulated and existing techniques for overcoming associated numerical instabilities and difficulties are discussed. These modules are then combined and solved to maximize bulk modulus and permeability in periodic materials with cubic elastic and isotropic flow symmetries. The multiphysics problem is formulated such that the final design is dependent on the relative importance, or weights, assigned by the designer to the competing stiffness and flow terms in the objective function. This allows the designer to tailor the microstructure according to the materials' future application, a feature clearly demonstrated by the presented results. The methodology can be extended to incorporate other material properties of interest as well as the design of composite materials.
SUMMARYWe propose the level set method of topology optimization as a viable, robust and efficient alternative to density-based approaches in the setting of fluid flow. The proposed algorithm maintains the discrete nature of the optimization problem throughout the optimization process, leading to significant advantages over density-based topology optimization algorithms. Specifically, the no-slip boundary condition is implemented directly-this is accurate, removes the need for interpolation schemes and continuation methods, and gives significant computational savings by only requiring flow to be modeled in fluid regions. Topological sensitivity information is utilized to give a robust algorithm in two dimensions and familiar two-dimensional power dissipation minimization problems are solved successfully. Computational efficiency of the algorithm is also clearly demonstrated on large-scale three-dimensional problems.
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