基于有限元的一类stefan问题数值模型研究

该文给出基于有限元方法的一类一维stefan问题的数值求解过程及算法.模型的建立基于已知的相变界面和固定边界处测得的温度和热流.模型的精度通过与Neumann获得的解析解的比较而得到验证.文中所讨论的模型可以用于反Stefan问题中自由边界的实时跟踪或者控制.最后,比较了已有的有限元模型,给出了仿真结果.     

A boundary element method for a multi-dimensional inverse heat conduction problem

In this paper, we investigate a variational method for a multi-dimensional inverse heat conduction problem in Lipschitz domains. We regularize the problem by using the boundary element method coupled with the conjugate gradient method. We prove the convergence of this scheme with and without Tikhonov regularization. Numerical examples are given to show the efficiency of the scheme.     

Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

ABSTRACT

Diffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable.     

2D inverse problem in a distributed parameter system

In this paper, we present a numerical solution for an inverse heat problem to estimate 2D space-wise coefficient in a parabolic system describing heat transfer. The inverse problem is recast into an optimization problem solved with a conjugate gradient method. This approach use the non linear optimization formulation by minimizing a cost function taking into account both the measured outputs of the system and the outputs calculated by means of the model. The solution is performed by using the free finite element software FreeFem. Numerical example is carried out to check the validity of the proposed method.     

一维热传导问题时变边界上热通量重构问题

具有Neumann边界条件的抛物方程的初边值问题是偏微分方程研究领域的一类经典问题.正问题是由已知的边界条件和初始条件来求区域温度场的问题.如果边界条件不足,但给出了区域内部的一些额外信息,这样便构成了一类热通量重构的反问题.本文讨论了一维热传导问题时动边界上的热通量重构问题,借助于位势理论方法,引入密度函数,将反问题本质上转化为一类关于密度函数的具有弱奇性核的第一类Volterra积分方程,采用了Tikhonov正则化,在正则化参数的选取上采用了后验的模型函数方法,数值结果验证了反演方法的有效性.     

On the numerical implementation of continuous adjoint sensitivity for transient heat conduction problems using an isogeometric approach

A crucial problem of continuous adjoint shape sensitivity analysis is the numerical implementation of its lengthy formulations. In this paper, the numerical implementation of continuous adjoint shape sensitivity analysis is presented for transient heat conduction problems using isogeometric analysis, which can serve as a tutorial guide for beginners. Using the adjoint boundary and loading conditions derived from the design objective and the primary state variable fields, the numerical analysis procedure of the adjoint problem, which is solved backward in time, is demonstrated. Following that, the numerical integration algorithm of the shape sensitivity using a boundary approach is provided. Adjoint shape sensitivity is studied with detailed explanations for two transient heat conduction problems to illustrate the numerical implementation aspects of the continuous adjoint method. These two problems can be used as benchmark problems for future studies.     

Identifying the parameters of elliptic-pseudoparabolic distributed systems

Calculation algorithms for the realization of gradient methods based on the solution to direct and adjoint problems in weak formulations are proposed for a number of inverse complex problems of estimating the parameters of multicomponent elliptic-pseudoparabolic distributed systems. The proposed approach makes it unnecessary to construct Lagrange functionals in explicit form and to use Green functions.     

基于最大期望算法的电阻抗成像系统的图像重建

电阻抗成像EIT(Electrical impedance tomography)技术利用不同媒质具有不同的电导率这一物理基础,通过测量目标场在一定电刺激下所呈现出的电特性,推导出目标场内部的电导率分布信息,进而推知该场中媒质的分布情况。EIT图像重建问题是一个非线性的病态逆问题,且测量系统往往存在噪声,使重建图像中存在伪影,传统的正则化方法对重建图像伪影的抑制能力有限。本文将一种统计学方法,即最大期望EM(expectation maximization)算法应用于EIT逆问题求解。它将EIT的数学模型转化为非负约束极小化问题,并通过梯度投影简化牛顿算法GPRN(gradient projection-reduced Newton iteration method)求解该问题。与传统的Tikhonov算法和共轭梯度算法CG(conjugate gradient)相比,有效地抑制了重建图像中伪影的产生。仿真和实验结果表明,EIT系统可以通过EM算法获得高质量的重建图像。     

Aerodynamic Shape Optimization Using First and Second Order Adjoint and Direct Approaches

This paper focuses on discrete and continuous adjoint approaches and direct differentiation methods that can efficiently be used in aerodynamic shape optimization problems. The advantage of the adjoint approach is the computation of the gradient of the objective function at cost which does not depend upon the number of design variables. An extra advantage of the formulation presented below, for the computation of either first or second order sensitivities, is that the resulting sensitivity expressions are free of field integrals even if the objective function is a field integral. This is demonstrated using three possible objective functions for use in internal aerodynamic problems; the first objective is for inverse design problems where a target pressure distribution along the solid walls must be reproduced; the other two quantify viscous losses in duct or cascade flows, cast as either the reduction in total pressure between the inlet and outlet or the field integral of entropy generation. From the mathematical point of view, the three functions are defined over different parts of the domain or its boundaries, and this strongly affects the adjoint formulation. In the second part of this paper, the same discrete and continuous adjoint formulations are combined with direct differentiation methods to compute the Hessian matrix of the objective function. Although the direct differentiation for the computation of the gradient is time consuming, it may support the adjoint method to calculate the exact Hessian matrix components with the minimum CPU cost. Since, however, the CPU cost is proportional to the number of design variables, a well performing optimization scheme, based on the exactly computed Hessian during the starting cycle and a quasi Newton (BFGS) scheme during the next cycles, is proposed.     

A method of solving the coefficient inverse problems of wave tomography

In this paper, the problem of nonlinear wave tomography is formulated as a coefficient inverse problem for a hyperbolic equation in the time domain. Efficient methods for solving the inverse problems of wave tomography for the case of transparent boundary conditions are presented. The algorithms are designed for supercomputers. We prove the Fréchet differentiability theorem for the residual functional and derive an exact expression for the Fréchet derivative in the case of a transparent boundary in the direct and conjugate problems. The expression for the Fréchet derivative of the residual functional remains valid if the experimental data are provided for only a part of the boundary. The effectiveness of the proposed method is illustrated by the numerical solution of a model problem of low-frequency wave tomography. The model problem is tailored to apply to the differential diagnosis of breast cancer.     

Constrained optimal control of Navier–Stokes flow by semismooth Newton methods

We propose and analyze a semismooth Newton-type method for the solution of a pointwise constrained optimal control problem governed by the time-dependent incompressible Navier–Stokes equations. The method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation. We analyze the flow control problem and prove q-superlinear convergence of the method. In the numerical implementation, adjoint techniques are combined with a truncated conjugate gradient method. Numerical results are presented that support our theoretical results and confirm the viability of the approach.     

Stochastic sensitivity analysis method for neural network learning

A new, efficient algorithm is developed for the sensitivity analysis of a class of continuous-time recurrent neural networks with additive noise signals. The algorithm is based on the stochastic sensitivity analysis method using the variational approach, and formal expressions are obtained for the functional derivative sensitivity coefficients. The present algorithm uses only the internal states and noise signals to compute the gradient information needed in the gradient descent method, where the evaluation of derivatives is not necessary. In particular, it does not require the solution of adjoint equations of the back-propagation type. Thus, the algorithm has the potential for efficiently learning the network weights with significantly fewer computations. The effectiveness of the algorithm in a statistical sense is shown, and the method is applied to the familiar layered network.     

建筑室内边界对流换热量的反向计算模型

针对目前反向计算模型还无法实现对建筑室内边界对流换热量进行反向计算这一制约性差距,采用温度贡献率方法,将边界对流换热量与室内测点温度之间表示成因果关系的温度贡献因子矩阵,基于计算流体力学,将最小二乘与Tikhonov正则化方法相结合,建立依据室内数个测点的离散温度求解边界对流换热量的反问题数学模型。应用三维通风空腔和某建筑内一间办公室进行实验验证,模型求解值与实测值的均方根差均小于80%,结果表明反向计算模型可以准确对室内边界对流换热量进行求解。     

Constrained optimization of nacelle shapes in Euler flow using semianalytical sensitivity analysis

The problem of aerodynamic shape optimization in Euler flow is addressed. B-splines are used for parametrization of the shape. Cheap gradient calculation is obtained via sensitivity analysis and the solution of an adjoint equation; pseudo secondorder spatial accuracy is achieved by means of a semianalytical formulation. As a validation of the approach, several inverse and constrained optimization test problems are presented with emphasis on civil engine nacelle design. The handling of nondifferentiable quantities (such as maxima) in cost functions is allowed for via the use of the Kreisselmeier-Steinhauser function.     

求解热传导系数反问题的量子行为粒子群算法

量子行为粒子群优化算法（QPSO）和Tikhonov正则化方法用来求解热传导反问题,近似估计平板随时间变化的热传导系数。由于热传导系数的函数形式是未知的,所以问题可以归结为函数估计问题。求解过程是基于最小二乘模型的,采用的是嵌在平板中的传感器所测量得到的温度,优化过程由QPSO算法来求解。给出了由L曲线方法选择正则参数的详细过程。提出算法的有效性经过了数值实验的验证。传感器的位置和数量对结果的影响也做了研究。给出了与共轭梯度法的比较。     

Continuous adjoint approach to the Spalart-Allmaras turbulence model for incompressible flows

A continuous adjoint formulation for the computation of the sensitivities of integral functions used in steady-flow, incompressible aerodynamics is presented. Unlike earlier continuous adjoint methods, this paper computes the adjoint to both the mean-flow and turbulence equations by overcoming the frequently made assumption that the variation in turbulent viscosity can be neglected. The development is based on the Spalart-Allmaras turbulence model, using the adjoint to the corresponding differential equation and boundary conditions. The proposed formulation is general and can be used with any other integral function. Here, the continuous adjoint method yielding the sensitivities of the total pressure loss functional for duct flows with respect to the normal displacements of the solid wall nodes is presented. Using three duct flow problems, it is demonstrated that the adjoint to the turbulence equations should be taken into account to compute the sensitivity derivatives of this functional with high accuracy. The so-computed derivatives almost coincide with “reference” sensitivities resulting from the computationally expensive direct differentiation. This is not, however, the case of the sensitivities computed without solving the turbulence adjoint equation, which deviates from the reference values. The role of all newly appearing terms in the adjoint equations, their boundary conditions and the gradient expression is investigated, significant and insignificant terms are identified and a study on the Reynolds number effect is included.     

Solution to boundary shape optimization problem of linear elastic continua with prescribed natural vibration mode shapes

This paper presents a practical method of numerical analysis for boundary shape optimization problems of linear elastic continua in which natural vibration modes approach prescribed modes on specified sub-boundaries. The shape gradient for the boundary shape optimization problem is evaluated with optimality conditions obtained by the adjoint variable method, the Lagrange multiplier method, and the formula for the material derivative. Reshaping is accomplished by the traction method, which has been proposed as a solution to boundary shape optimization problems of domains in which boundary value problems of partial differential equations are defined. The validity of the presented method is confirmed by numerical results of three-dimensional beam-like and plate-like continua.     

Solution of inverse boundary-value problems for multicomponent parabolic distributed systems

Computational algorithms implementing gradient methods based on solution of direct and adjoint problems in weak formulations are proposed for a number of complex-valued inverse problems of parameter renewal in multicomponent parabolic distributed systems. This approach makes it unnecessary to construct Lagrangian functionals explicitly and to use Green’s functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 49–73, July–August 2007.     

Parallel and Systolic Solution of Normalized Explicit Approximate Inverse Preconditioning

A new class of normalized approximate inverse matrix techniques, based on the concept of sparse normalized approximate factorization procedures are introduced for solving sparse linear systems derived from the finite difference discretization of partial differential equations. Normalized explicit preconditioned conjugate gradient type methods in conjunction with normalized approximate inverse matrix techniques are presented for the efficient solution of sparse linear systems. Theoretical results on the rate of convergence of the normalized explicit preconditioned conjugate gradient scheme and estimates of the required computational work are presented. Application of the new proposed methods on two dimensional initial/boundary value problems is discussed and numerical results are given. The parallel and systolic implementation of the dominant computational part is also investigated.     

A Self-Referencing Level-Set Method for Image Reconstruction from Sparse Fourier Samples

We address an ill-posed inverse problem of image estimation from sparse samples of its Fourier transform. The problem is formulated as joint estimation of the supports of unknown sparse objects in the image, and pixel values on these supports. The domain and the pixel values are alternately estimated using the level-set method and the conjugate gradient method, respectively. Our level-set evolution shows a unique switching behavior, which stabilizes the level-set evolution. Furthermore, the trade-off between the stability and the speed of evolution can be easily controlled by the number of the conjugate gradient steps, thus avoiding the re-initialization steps in conventional level set approaches.