23 research outputs found
A Subgradient Method for Free Material Design
A small improvement in the structure of the material could save the
manufactory a lot of money. The free material design can be formulated as an
optimization problem. However, due to its large scale, second-order methods
cannot solve the free material design problem in reasonable size. We formulate
the free material optimization (FMO) problem into a saddle-point form in which
the inverse of the stiffness matrix A(E) in the constraint is eliminated. The
size of A(E) is generally large, denoted as N by N. This is the first
formulation of FMO without A(E). We apply the primal-dual subgradient method
[17] to solve the restricted saddle-point formula. This is the first
gradient-type method for FMO. Each iteration of our algorithm takes a total of
foating-point operations and an auxiliary vector storage of size O(N),
compared with formulations having the inverse of A(E) which requires
arithmetic operations and an auxiliary vector storage of size . To
solve the problem, we developed a closed-form solution to a semidefinite least
squares problem and an efficient parameter update scheme for the gradient
method, which are included in the appendix. We also approximate a solution to
the bounded Lagrangian dual problem. The problem is decomposed into small
problems each only having an unknown of k by k (k = 3 or 6) matrix, and can be
solved in parallel. The iteration bound of our algorithm is optimal for general
subgradient scheme. Finally we present promising numerical results.Comment: SIAM Journal on Optimization (accepted
Total variation regularization of multi-material topology optimization
This work is concerned with the determination of the diffusion coefficient
from distributed data of the state. This problem is related to homogenization
theory on the one hand and to regularization theory on the other hand. An
approach is proposed which involves total variation regularization combined
with a suitably chosen cost functional that promotes the diffusion coefficient
assuming prespecified values at each point of the domain. The main difficulty
lies in the delicate functional-analytic structure of the resulting
nondifferentiable optimization problem with pointwise constraints for functions
of bounded variation, which makes the derivation of useful pointwise optimality
conditions challenging. To cope with this difficulty, a novel reparametrization
technique is introduced. Numerical examples using a regularized semismooth
Newton method illustrate the structure of the obtained diffusion coefficient.
The Free Material Design problem for stationary heat equation on low dimensional structures
For a given balanced distribution of heat sources and sinks, , we find an
optimal conductivity tensor field, , minimizing the thermal compliance.
We present in a rather explicit form in terms of the datum. Our
solution is in a cone of non-negative tensor-valued finite Borel measures. We
present a series of examples with explicit solutions.49J20, %Singular parabolic
equations secondary: 49K20, 80M5