Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media

We are given a uniformly elliptic coefficient field that we regard as a realization of a stationary and finite-range ensemble of coefficient fields. Given a right-hand side supported in a ball of size $$\ell \gg 1$$ and of vanishing average, we are interested in an algorithm to compute the solution near the origin, just using the knowledge of the given realization of the coefficient field in some large box of size $$L\gg \ell $$ . More precisely, we are interested in the most seamless artificial boundary condition on the boundary of the computational domain of size L. Motivated by the recently introduced multipole expansion in random media, we propose an algorithm. We rigorously establish an error estimate on the level of the gradient in terms of $$L\gg \ell \gg 1$$ , using recent results in quantitative stochastic homogenization. More precisely, our error estimate has an a priori and an a posteriori aspect: with a priori overwhelming probability, the prefactor can be bounded by a constant that is computable without much further effort, on the basis of the given realization in the box of size L. We also rigorously establish that the order of the error estimate in both L and $$\ell $$ is optimal, where in this paper we focus on the case of $$d=2$$ . This amounts to a lower bound on the variance of the quantity of interest when conditioned on the coefficients inside the computational domain, and relies on the deterministic insight that a sensitivity analysis with respect to a defect commutes with stochastic homogenization. Finally, we carry out numerical experiments that show that this optimal convergence rate already sets in at only moderately large L, and that more naive boundary conditions perform worse both in terms of rate and prefactor.

Selection of optimal artificial boundary condition (ABC) frequencies for structural damage identification

We determine optimal inflow boundary perturbations to steady flow through a straight inflexible tube with a smooth axisymmetric stenosis at a bulk-flow Reynolds number $\mathit{Re}= 400$, for which the flow is asymptotically stable. The perturbations computed produce an optimal gain, i.e. kinetic energy in the domain at a given time horizon normalized by a measure of time-integrated energy on the inflow boundary segment. We demonstrate that similarly to the optimal initial condition problem, the gain can be interpreted as the leading singular value of the forward linearized operator that evolves the boundary conditions to the final state at a fixed time. In this investigation we restrict our attention to problems where the temporal profile of the perturbations examined is a product of a Gaussian bell and a sinusoid, whose frequency is selected to excite axial wavelengths similar to those of the optimal initial perturbations in the same geometry. Comparison of the final state induced by the optimal boundary perturbation with that induced by the optimal initial condition demonstrates a close agreement for the selected problem. Previous works dealing with optimal boundary perturbation considered a prescribed spatial structure and computed an optimal temporal variation of a wall-normal velocity component, whereas in this paper we consider the problem of a prescribed temporal structure and compute the optimal spatial variation of velocity boundary conditions over a one-dimensional inflow boundary segment. The methodology is capable of optimizing boundary perturbations in general non-parallel two- and three-dimensional flows.

Existing Poisson mesh editing techniques mainly focus on designing schemes to propagate deformation from a given boundary condition to a region of interest. Although solving the Poisson system in the least-squares sense distributes the distortion errors over the entire region of interest, large deformation in the boundary condition might still lead to severely distorted results. We propose to optimize the boundary condition (the merging boundary) for Poisson mesh merging. The user needs only to casually mark a source region and a target region. Our algorithm automatically searches for an optimal boundary condition within the marked regions such that the change of the found boundary during merging is minimal in terms of similarity transformation. Experimental results demonstrate that our merging tool is easy to use and produces visually better merging results than unoptimized techniques.

Absorbing boundary conditions are widely used in numerical modeling of wave propagation in unbounded media to reduce reflections from artificial boundaries (Lindman, 1975; Clayton and Engquist, 1977; Reynolds, 1978; Liao et al., 1984; Cerjan et al., 1985; Randall, 1988; Higdon, 1991). We are interested in a particular absorbing boundary condition that has maximum absorbing ability with a minimum amount of computation and storage. This is practical for 3-D simulation of elastic wave propagation by a finite‐difference method. Peng and Toksöz (1994) developed a method to design a class of optimal absorbing boundary conditions for a given operator length. In this short note, we give a brief introduction to this technique, and we compare the optimal absorbing boundary conditions against those by Reynolds (1978) and Higdon (1991) using examples of 3-D elastic finite‐difference modeling on an nCUBE-2 parallel computer. In the Appendix, we also give explicit formulas for computing coefficients of the optimal absorbing boundary conditions.

A reverse computation based on adjoint formulation of forced convection heat transfer is proposed to obtain the optimal thermal boundary conditions for heat transfer characteristics; for example, a total heat transfer rate or a temperature at a specific location. In the reverse analysis via adjoint formulation, the heat flow is reversed in both time and space. Thus, using the numerical solution of the adjoint problem, we can inversely predict the boundary condition effects on the heat transfer characteristics. As a result, we can obtain the optimal thermal boundary conditions in both time and space to control the heat transfer at any given time. © 2004 Wiley Periodicals, Inc. Heat Trans Asian Res, 33(3): 161–174, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20002

A reverse computation based on adjoint formulation of forced convection heat transfer is proposed to obtain the optimal thermal boundary conditions for heat transfer characteristics ; such as a total heat transfer rate or a temperature at a specific location. In the reverse analysis via adjoint formulation, the heat flow is reversed in both time and space. Thus, using the numerical solution of the adjoint problem, we can inversely predict the boundary condition effects on the heat transfer characteristics. As a result, we can obtain the optimal thermal boundary conditions in both time and space to control the heat transfer at any given time.

The non-overlapping Schwarz method with absorbing boundary conditions instead of the Dirichlet boundary conditions is an efficient variant of the overlapping Schwarz method for the Helmholtz equation. These absorbing boundary conditions defined on the interface between the subdomains are the key ingredients to obtain a fast convergence of the iterative Schwarz algorithm. In a one-way subdomains splitting, non-local optimal absorbing boundary conditions can be obtained and leads to the convergence of the Schwarz algorithm in a number of iterations equal to the number of subdomains minus one. This paper investigates different local approximations of these optimal absorbing boundary conditions for finite element computations in acoustics. Different approaches are presented both in the continuous and in the discrete analysis, including high-order optimized continuous absorbing boundary conditions, and discrete absorbing boundary conditions based on algebraic approximation. A wide range of new numerical experiments performed on unbounded acoustics problems demonstrate the comparative performance and the robustness of the proposed methods on general unstructured mesh partitioning.

Contributing up to 40~70% of total errors deteriorating the machining quality of machine tools, thermal errors need to be decreased. Building the accurate thermal simulation model of spindle system is the premise to study its thermal characteristics and further reduce its thermal errors. However, the accuracy of the thermal simulation model mainly rests with the accuracy of the boundary conditions since the mesh model is accurate enough. In overcoming the poor consistency between the thermal experiment and thermal simulation, thus obtaining the accurate thermal simulation model, the empirically calculated boundary conditions should be optimized, which is essentially an inverse problem. Treating the heat generation rates and the convective heat transfer coefficients as the optimization objects, and the simulation difference as the objective function, a dynamic metamodel assisted differential evolution (DMDE) method is adopted to efficiently search the optimal boundary conditions. The metamodel generated on the already executed data can prescreen out the most promising trial data to speed up the convergence of the differential evolution. And the scoring strategy is used to select the top high possible trial data to further accelerate the convergence. Results demonstrate that the optimal boundary conditions can be obtained using this method with maximum temperature simulation error reduced from 85.6 to 4.9% and thermal extension simulation error reduced from 60.9 to− 3.5%. Furthermore, the number of executions of thermal simulation analysis is reduced from 2000 needed by genetic algorithm or differential evolution to an average of 206 needed by the adopted method.

This paper presents an optimal absorbing boundary condition designed to model acoustic and elastic wave propagation in two-dimensional and three-dimensional media using the finite difference method. In this condition, extrapolation on the artificial boundaries of a finite difference domain is expressed as a linear combination of wave fields at previous time steps and/or interior grids. The acoustic and elastic reflection coefficients from the artificial boundaries are derived. They are found to be identical to the transfer functions of two cascaded systems—the inverse of a causal system and an anticausal system. The method makes use of the zeros and poles of reflection coefficients in a complex plane. The optimal absorbing boundary condition described in this paper yields, on the average, reflection coefficients about 10-dB smaller than Higdon’s absorbing boundary condition, and about 20-dB smaller than Reynolds’ absorbing boundary condition.

Modified electromyography-assisted optimization approach for predicting lumbar spine loading while walking with backpack loads.

Approximate effective coefficients of random heterogeneous materials could be obtained by solving auxiliary problems with certain boundary condition. When the coefficients of the auxiliary problems have great jump caused by the random heterogeneous materials with high-contrast properties, Dirichlet boundary condition (DBC) and Neumann boundary condition (NBC) provide broad upper and lower bounds of effective coefficients, respectively. Two possible factors that result in inaccurate approximations of effective coefficients are discussed in this paper. Effect of large condition number of stiffness matrix caused by the high contrast on the numerical accuracy of approximate effective coefficients is analysed. Since DBC and NBC are not effective for the high-contrast materials, an alternative Robin boundary condition (RBC) is presented to provide much better approximations of effective coefficients. Convergence of the approximate effective coefficients under RBC is proved. Numerical examples indicate that proper adjusting factor introduced in RBC makes it more flexible than other boundary conditions. RBC is more suitable for the high-contrast materials and has potential to be an optimal boundary condition.

One of the methods commonly used to numerically solve a problem in an infinite domain is the method of artificial boundary conditions [1]. For a linear scalar problem, this method may be summarized as follows:

In a three-dimensional domain Ω with J cylindrical outlets to infinity the problem is treated how solutions to the stationary Stokes and Navier–Stokes system with pressure conditions at infinity can be approximated by solutions on bounded subdomains. The optimal artificial boundary conditions turn out to have singular coefficients. Existence, uniqueness and asymptotically precise estimates for the truncation error are proved for the linear problem and for the nonlinear problem with small data. The results include also estimates for the so called “do-nothing” condition.

A 4D-Var data assimilation technique is applied to the rectangular-box configuration of the NEMO in order to identify the optimal parametrization of boundary conditions at lateral boundaries. The case of the staircase-shaped coastlines is studied by rotating the model grid around the center of the box. It is shown that, in some cases, the formulation of the boundary conditions at the exact boundary leads to appearance of exponentially growing modes while optimal boundary conditions allow to correct the errors induced by the staircase-like appriximation of the coastline.