Optimization Algorithms for Hierarchical Problems with Application to Nanoporous Materials

Abstract

We present optimization algorithms for the design of complex hierarchical systems, motivated by applications to the design of nanoporous materials. Nanoporous materials have a broad range of engineering applications, including gas storage and filtration, electrical energy storage in batteries and capacitors, and catalysis. The design of such materials involves modeling of the material over many length scales, leading to a hierarchy of mathematical models. Our algorithms are also hierarchical in structure with the goal of exploiting the model hierarchy to obtain solutions more rapidly. We discuss the choice of optimization models, initialization schemes, the hierarchical optimization algorithm, software design, and computational results. Introduction.We present optimization algorithms for the design of complex hierarchical systems, motivated by applications to the design of nanoporous materials. Nanoporous materials have a broad range of engineering applications, including gas storage and filtration, electrical energy storage in batteries and capacitors, and catalysis [1,18,36,8,29,21]. Because the design of such materials involves modeling of the material over many length scales, the underlying mathematical models can be considered as a hierarchy of models. Our algorithms are also hierarchical in structure with the goal of exploiting the model hierarchy to obtain solutions more rapidly.An important feature of nanoporous materials is that within the nanopores, a gas can be stored at a much higher density than is possible with equal pressure in an open tank. The problem for applications is that the diffusion of gas into and out of the storage material is very slow. To speed this process, engineers have considered putting channels of various widths into the material; this can be thought of as adding a "vascular" system. The trade-off is, of course, between speeding up the filling or emptying of the material and storing as much as possible. One version of the optimization problem, then, is to maximize the amount of gas that can be stored and extracted in a timely manner subject to sacrificing a fixed amount of volume that is made available for transport channels. Beyond storage of gas, nanoporous materials afford analogous benefits in the storage of electric charge in electrochemical capacitors and batteries.

Using Discrete and Continuous Models to Solve Nanoporous Flow Optimization Problems

“…The problem and the hierarchical optimization algorithm are described in [2]. The focus here is on the use of the discrete model within that algorithm.…”

“…Many important applications, including the storage of electric charge, are noted in [2]. The problem is that it is difficult to charge, i.e., fill, these materials rapidly.…”

“…As noted above, our approach to solving these problems is described in [2]. Briefly, we specify a PDE on the flow's domain and allow channels to be built along the edges of the finite element grid.…”

“…The problem on the finest scale captures all of the variables of the optimization problem and is where the ultimate problem is formulated. This problem, however, can be huge; exploiting this hierarchical structure, as shown in [2], can lead to significant computational savings.…”

“…The problem and the hierarchical optimization algorithm are described in [2]. A key feature of the model is the fact that we use the edges of the finite element grid as the locations of the channels.…”