10 research outputs found
An efficient threshold dynamics method for topology optimization for fluids
We propose an efficient threshold dynamics method for topology optimization
for fluids modeled with the Stokes equation. The proposed algorithm is based on
minimization of an objective energy function that consists of the dissipation
power in the fluid and the perimeter approximated by nonlocal energy, subject
to a fluid volume constraint and the incompressibility condition. We show that
the minimization problem can be solved with an iterative scheme in which the
Stokes equation is approximated by a Brinkman equation. The indicator functions
of the fluid-solid regions are then updated according to simple convolutions
followed by a thresholding step. We demonstrate mathematically that the
iterative algorithm has the total energy decaying property. The proposed
algorithm is simple and easy to implement. A simple adaptive time strategy is
also used to accelerate the convergence of the iteration. Extensive numerical
experiments in both two and three dimensions show that the proposed iteration
algorithm converges in much fewer iterations and is more efficient than many
existing methods. In addition, the numerical results show that the algorithm is
very robust and insensitive to the initial guess and the parameters in the
model.Comment: 23 pages, 24 figure
A prediction-correction based iterative convolution-thresholding method for topology optimization of heat transfer problems
In this paper, we propose an iterative convolution-thresholding method (ICTM)
based on prediction-correction for solving the topology optimization problem in
steady-state heat transfer equations. The problem is formulated as a
constrained minimization problem of the complementary energy, incorporating a
perimeter/surface-area regularization term, while satisfying a steady-state
heat transfer equation. The decision variables of the optimization problem
represent the domains of different materials and are represented by indicator
functions. The perimeter/surface-area term of the domain is approximated using
Gaussian kernel convolution with indicator functions. In each iteration, the
indicator function is updated using a prediction-correction approach. The
prediction step is based on the variation of the objective functional by
imposing the constraints, while the correction step ensures the monotonically
decreasing behavior of the objective functional. Numerical results demonstrate
the efficiency and robustness of our proposed method, particularly when
compared to classical approaches based on the ICTM.Comment: 29 pages, 25 figure
Escape times for subgraph detection and graph partitioning
We provide a rearrangement based algorithm for fast detection of subgraphs of
vertices with long escape times for directed or undirected networks.
Complementing other notions of densest subgraphs and graph cuts, our method is
based on the mean hitting time required for a random walker to leave a
designated set and hit the complement. We provide a new relaxation of this
notion of hitting time on a given subgraph and use that relaxation to construct
a fast subgraph detection algorithm and a generalization to -partitioning
schemes. Using a modification of the subgraph detector on each component, we
propose a graph partitioner that identifies regions where random walks live for
comparably large times. Importantly, our method implicitly respects the
directed nature of the data for directed graphs while also being applicable to
undirected graphs. We apply the partitioning method for community detection to
a large class of model and real-world data sets.Comment: 22 pages, 10 figures, 1 table, comments welcome!
Recent Advances in Industrial and Applied Mathematics
This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress
Recent Advances in Industrial and Applied Mathematics
This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress