Finite-element approximation of 2D elliptic optimal design

A class of optimal shape design problems is studied in this paper. The boundary shape of a domain is determined such that the solution of the underlying partial differential equation matches, as well as possible, a given desired state. In the original optimal shape design problem, the variable domain is parameterized by a class of functions in such a way that the optimal design problem is changed to an optimal control problem on a fixed domain. Then, the resulting distributed control problem is embedded in a measure theoretical form, in fact, an infinite-dimensional linear programming problem. The optimal measure representing the optimal shape is approximated by a solution of a finite-dimensional linear programming problem. The method is evaluated via a numerical example.

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. Šveràk in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.

This paper is concerned with an optimal shape design problem in aerodynamics. The inverse problem in question consists in finding the optimal shape an airfoil placed in a potential flow at a given angle of attack should have such that the pressure distribution on its surface matches a desired one. The numerical method to achieve this aim is based on a body-fitted grid generation technique (elliptic, O-type) to generate a mesh over the airfoil surface and solve for the flow equation. The O-type scheme is used due to its ability to generate a high quality (fine and orthogonal) grid around the airfoil surface. This paper describes a novel and very efficient sensitivity analysis scheme to compute the sensitivity of the pressure distribution to variation of grid node positions and both the conjugate gradient method (CGM) and a version of the quasi-Newton method (i.e., BFGS) are used as optimization algorithms to minimize the difference between the computed pressure distribution on the airfoil surface and desired one. The elliptic grid generation technique allows us to map the physical domain (body) onto a fixed computational domain and to discretize the flow equation using the finite difference method (FDM).

The problem of optimal design for quasi-brittle elastic bodies with cracks is considered in cases of static and cyclic loading. Optimization problems considered consist of rinding the boundary of a body such that the optimized functional (volume of the body) attains an extremal value, while satisfying prescribed bounds on stress intensity factors and minimum number of cycles (structural durability). Such problems are characterized by incomplete information concerning crack size, location, and orientation. In this context, the paper presents some possible formulations of optimal structural design problems based on minimax (guaranteed) approaches. Using these approaches, optimal designs are obtained for beams and plates with surface and internal cracks.

Nanoparticles optimal shape designing in the cylindrical coordinate to use for optical devices

A general approach to [formula omitted]-optimal designs for weighted univariate polynomial regression models

The two papers in this issue share a common concern: smoothing. In the first paper, it is time series that are being smoothed, and in the second paper it is multigrid methods that do the smoothing for PDE-constrained optimization problems. In biology, medicine, finance, economics, geophysics, and social sciences one frequently needs to discern “trends” in data sets. Mathematically, this problem is known as filtering, smoothing, or time series analysis, and many different methods have been devised, including moving average filters, bandpass filters, and smoothing splines. For a given set of scalars $y_t$, one wants to find another set of scalars $x_t$ that vary smoothly, but that are close to the original set. The new points $x_t$ are considered to represent the underlying trend. The question now is, how do we define what it means to “vary smoothly”? Seung-Jean Kim, Kwangmoo Koh, Stephen Boyd, and Dimitry Gorinevsky answer this question in the one-norm. In their paper “$\ell_1$ Trend Filtering,” they minimize an expression containing a one-norm, so that the resulting $x_t$ are points of a piecewise linear function. The minimization is formulated as a convex quadratic program and solved by an interior point method. This paper should be of interest to many readers, because of its connections to $\ell_1$ regularization in geophysics, signal processing, statistics, and sparse approximation. In their paper “Multigrid Methods for PDE Optimization,” Alfio Borzì and Volker Schulz review multigrid methods for solving infinite-dimensional optimization problems whose constraints are expressed in terms of partial differential equations (PDEs). Such optimization problem arise in optimal control, shape design, and shape optimization. Multigrid methods, very informally, solve PDEs by discretizing them iteratively on a hierarchy of grids, so as to capture all frequencies. So-called smoothers are responsible for high frequencies, while lower frequencies are resolved on coarser grids. This paper is essentially self-contained. It starts by introducing terminology for PDE-constrained optimization problems and reviewing multigrid methods. Subsequent discussions focus on multigrid SQP, Schur-complement-based multigrid smoothers, and collective smoothing multigrid, as well as applications to optimal control problems governed by hyperbolic, elliptic, and parabolic PDEs.

The photovoltaic/thermal (PV/T) system can be integrated with other energy resources and, the optimal design of such integrated PV/T systems becomes important. Also, fuel cell with hydrogen consumption is significantly useful in combination with other renewable energy systems. The objective of this study is to investigate the effects of PV/T and fuel cell as a combined heat and power system on optimal design problem with considerations for PV/T unreliability. The contributions of this study are consideration unreliability of PV/T panels output heat and power and investigation the effects of PV/T and fuel cell on optimal design problem are investigated. Some limitations such as battery size, PV/T installation area, and demand unreliability are subjected to problem. As heuristic algorithm is applied in optimal design problem, it is shown that utilization of PV/T reduces cost by 1.10%. Also, with utilization of fuel cell beside other units, total cost is reduced as compared with literature. A critical sensitivity analysis is done and, it is concluded that several factors, namely, PV/T, energy storage system, conventional distributed generation, fuel cell, fuel prices, and power and heat profiles have considerable effects on optimal design. Among them, battery and fuel cell prices have the most and least effects.

The goal of optimal shape design problems is to find a shape of the structure, belonging to an apriori defined class U ad and minimizing a cost functional J on U ad . Shape optimization of systems, governed by equations is now more or less standard. On the other hand there exist a lot of problems where the mathematical model leads to the so called variational inequality. It is the aim of this contribution to extend optimal shape design problems to systems described by variational inequalities. It is well known that the optimal control of such systems has one specific feature, namely the problem is non-smooth. Here we deal with one of the most typical applications of variational inequalities in solid mechanics, namely with the so called contact problems for linearly elastic bodies. A detailed mathematical analysis can be found in [3].KeywordsVariational InequalityContact ForceContact ProblemElastic BodyOptimal ShapeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, the author presents some results concerning the optimal shape design problem gov erned by the variational inequalities. The optimization techniques used for the solutions and present optimality conditions are also discussed. The computations are done by finite element methods, and the performance criterion is minimized by the gradient method. Some examples are used to clearly illustrate the essential elements of shape optimal design problems.

The selection of items from a calibrated item bank for fixed-form tests is an optimal test design problem; this problem has been handled in the literature by mathematical programming models. A similar problem, however, arises when items are selected for an adaptive test or for testlets. This paper focuses on the similarities of optimal design of fixed-form tests, adaptive tests, and testlets within the framework of the general theory of optimal designs. A sequential design procedure is proposed that uses these similarities. This procedure not only enables optimal design of fixed-form tests, adaptive tests, and testlets, but is also very flexible. The procedure is easy to apply, and consistent estimates for the trait level distribution are obtained. Index terms: adaptive tests, consistency, efficiency, optimal test design, sequential procedure, test design, testlets.

The refinement of the global aerodynamical optimal design (OD) of the shape of a flying configuration (FC) can be performed by improving of the start solutions for the optimization and/or of the optimization strategy itself. The proposed strategy is the own developed iterative optimum–optimorum theory. The study is here focused on the further improvement of her new, original, reinforced, zonal, spectral solutions for the partial–differential equations (PDEs) of the three–dimensional compressible Navier–Stokes layer (NSL), which govern the flow over the FCs, in subsonic and supersonic flow. These NSL's solutions, which are good suited for the computation and, especially, for the global optimal design, use the analytical potential solutions of the flow over the same FC twice: firstly as outer flow, at the NSL's edge (instead of the parallel flow used by Prandtl in his boundary layer theory) and, secondly, the velocity's components are products between the corresponding potential velocities and polynomial expansions with arbitrary coefficients, which are used to satisfy the NSL's PDEs. The use of analytical elliptical potential solutions leads to subsonic and the use of hyperbolical potential solutions leads to supersonic stabilized and rapid convergent NSL's solutions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modeling private highways in networks with entry–exit based toll charges

A unified method for continuum shape design sensitivity analysis and optimal design of mechanical components is developed. A domain method of shape design sensitivity analysis that uses the material derivative concept of continuum mechanics is employed. For numerical implementation of shape optimal design, parameterization of the boundary shape of mechanical components is defined and illustrated using a Bezier surface. In shape design problems, nodal points of the finite element model move as the shape changes. A method of automatic regridding to account for shape change has been developed using a design velocity field in the physical domain that obeys the governing equilibrium equations of the elastic solid. For numerical implementation of the continuum shape design sensitivity analysis and automatic regridding, an established finite element analysis code is used. To demonstrate the feasibility of the method developed, shape design optimization of a main engine bearing cap is carried out as an example.

Because of their superior mechanical, structural, and electronic properties, graphene sheets (GSs) are supposed to be components of nanoelectromechanical systems (NEMS). In this study, we carry out shape and structural optimization of GSs with topological defects in natural vibration problem to enhance their fundamental frequencies. At first, we model GSs as continuum frame models based on the molecular mechanics method. Then, we carry out shape design optimization of GSs by using a developed free-form optimization method for frame structures. In the shape design optimization, we use the fundamental frequency as objective function and enhance it considering volume constraint and repeated eigenvalues. Next, we derive the optimal atom structures of GSs with topological defects using a combination of the Phase-Field-Crystal method, the Voronoi tessellation method, and molecular dynamics simulation. The numerical results show that the fundamental frequencies of GSs can be significantly increased according to the shape and structural design optimization, and we can get their optimal atom structures with defects, which is helpful for designing GSs as components in NEMS.