Modelling and estimation for a moving boundary problem

The displacement of a slightly compressible liquid by another in a porous medium has been modelled. This problem, which involves a moving boundary, has been numerically solved for the one-dimensional case by using the Galerkin method. State and parameter estimation have been carried out using the extended Kalman filter.     

The boundary element method applied to a moving free boundary problem

In this paper a boundary problem is considered for which the boundary is to be determined as part of the solution. A time‐dependent problem involving linear diffusion in two spatial dimensions which results in a moving free boundary is posed. The fundamental solution is introduced and Green’s Theorem is used to yield a non‐linear system of integral equations for the unknown solution and the location of the boundary. The boundary element method is used to obtain a numerical solution to this system of integral equations which in turn is used to obtain the solution of the original problem. Graphical results for a two‐dimensional problem are presented. Published in 1999 by John Wiley & Sons, Ltd.     

A numerical simulation for two-dimensional moving boundary problems with a mushy zone

For materials such as alloy, organic phase-change materials and many others, the change of phases may take place over a temperature range. This leads to phase-change problems with the mushy zone in which the solid and liquid phases coexist. The present study introduces a numerical method combining the Laplace transform technique and the control volume method to solve two-dimensional phase-change problems with the mushy zone. The hybrid numerical method involves the control volume formulation for the space domain and the Laplace transform technique for the time domain. The Taylor's series approximation is applied to linearize nonlinear terms in the governing equation. The transfinite mapping method is used to generate control-volume meshes in each region. The growth of the mushy zone is unknown a priori and is predicted by using the least-square iteration scheme. It will be found that the present hybrid numerical method can be efficiently applied to solve two-dimensional phase-change problems with a mushy zone.     

The singular function boundary integral method for a two-dimensional fracture problem

The singular function boundary integral method (SFBIM) originally developed for Laplacian problems with boundary singularities is extended for solving two-dimensional fracture problems formulated in terms of the Airy stress function. Our goal is the accurate, direct computation of the associated stress intensity factors, which appear as coefficients in the asymptotic expansion of the solution near the crack tip. In the SFBIM, the leading terms of the asymptotic solution are used to approximate the solution and to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of Green's theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The numerical results on a model problem show that the method converges extremely fast and yields accurate estimates of the leading stress intensity factors.     

Aggregated state dynamic programming for a multiobjective two-dimensional bin packing problem

This paper studies a real-life multi-objective two-dimensional single-bin-size bin-packing problem arising in industry. A packing pattern is defined by one bin, a set of items packed into the bin and the packing positions of these items. A number of bins can be placed with the same packing pattern. The objective is not only to minimise the number of bins used, as in traditional bin-packing problems, but also to minimise the number of packing patterns. Based on our previous study of a heuristic stemming from dynamic programming by aggregating states to avoid the exponential increase in the number of states, we further develop this heuristic by decomposing a pattern with a number of bins at each step. Computational results show that this heuristic provides satisfactory results with a gap generally less than 20% with respect to the optimum.     

Solution of a drying problem for a colloidal porous material with a moving evaporation boundary

Closed solutions are derived for linear differential equations for one-dimensional drying with an evaporation boundary moving in accordance witht.Translated from Inzhenerno-Fizieheskii Zhurnal, Vol. 25, No. 5, pp. 871–876, November, 1973.     

An efficient dual boundary element technique for a two-dimensional fracture problem with multiple cracks

An efficient dual boundary element technique for the analysis of a two-dimensional finite body with multiple cracks is established. In addition to the displacement integral equation derived for the outer boundary, since the relative displacement of the crack surfaces is adopted in the formulation, only the traction integral equation is established on one of the crack surfaces. For each crack, a virtual boundary is devised and connected to one of the crack surfaces to construct a closed integral path. The rigid body translation for the domain enclosed by the closed integral path is then employed for evaluating the hypersingular integral. To solve the dual displacement/traction integral equations simultaneously, the constant and quadratic isoparametric elements are taken to discretize the closed integral paths/crack surfaces and the outer boundary, respectively. The present method has distinct computational advantages in solving a fracture problem which has arbitrary numbers, distributions, orientations and shapes of cracks by a few boundary elements. Several examples are analysed and the computed results are in excellent agreement with other analytical or numerical solutions.     

Algorithms and results of solving the inverse heat-conduction boundary problem in a two-dimensional formulation

The effectiveness of gradient algorithms for solving the inverse problem which are regulated in terms of the number of iterations is investigated.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 48, No. 4, pp. 658–666, April, 1985.     

A well-conditioned technique for solving the inverse problem of boundary traction estimation for a constrained vibrating structure

 Estimation of the frequency and spatial dependent boundary traction vector from measured vibration responses in a vibrating structure is addressed. This problem, also referred to as the inverse problem, may in some circumstances be ill-conditioned. Here a technique to overcome the ill-conditioning is proposed. A subset of a set of available eigenmodes is chosen such that the problem becomes well-conditioned enough. It is shown that the ill-conditioning originates from the fact that not all eigenmodes are orthogonal over the surface where the traction vector is sought. Consequently, by choosing a set of eigenmodes orthogonal over the surface of interest, the problem becomes well-conditioned. The calculated traction vector is shown to converge to the true one in the sense of a L₂-norm on the boundary of the body. The proposed technique is verified, using numerical simulation of measured responses, with good agreement. Received: 10 January 2002 / Accepted: 24 October 2002 This work was performed under contract from the Swedish Defence Material Administration (FMV). The funding provided is gratefully acknowledged. The author would like to thank all staff at the Structural Dynamics research group, Department of Structures and Materials, Aeronautics Division, Swedish Defence Research Agency.     

Modified boundary conditions for two-dimensional gasdynamic calculations in regions of arbitrary shape with moving boundaries present

Boundary conditions enabling one to improve the accuracy, convergence, and economy on numerical calculations are discussed.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 50, No. 1, pp. 40–48, January, 1986.     

Some moving boundary problems for a generalised diffusion equation

Diffusion controlled growth of planar, cylindrical or spherical particles is considered where the composition satisfies a generalised diffusion equation. Asymptotic methods are used to determine composition gradients near the growing particles in the cases of “fast” or “slow” growth. It is shown that in these limits under certain conditions a constant average diffusion coefficient can be defined for which the growth rate calculated from the differential equation with this constant diffusion coefficient agrees with that with the full variable coefficient. Similar considerations are also made of the case of composition dependent diffusion and bulk flow for growing spherical particles.     

Asymptotic representation of the solution of the first boundary-value problem for the heat-conduction equation with a moving boundary

We obtain an asymptotic solution of the first boundary-value problem for the heat-conduction equation in a region with a moving boundary, which initially may degenerate to a point.     

Solution of a two-dimensional heat-conduction problem for a sector

We consider a problem for the Laplace equation in a circular sector wherein heat exchange takes place on the sides of the sector in accordance with Newton's law and a temperature distribution is specified on the circular arc.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 54, No. 1, pp. 133–139, January, 1988.     

A fixed domain method for diffusion with a moving boundary

Summary The one-dimensional diffusion equation for a region with one fixed boundary and one unknown moving boundary is transformed to a non-linear equation on a fixed region by using the moving boundary position as the time variable. The boundary velocity becomes a second dependent variable, with dependence only on the new time variable. An implicit finite difference scheme, marching in time, is applied to a problem with known analytic solution to demonstrate the computing speed and accuracy of this approach, and also to a problem solved previously by variable time step methods. This transformation reduces any parabolic or elliptic system of equations on a domain with moving boundary, or with unknown free surface in two space variables, to a non-linear fixed domain system which has advantages for computation.     

A boundary integral equation method for a Neumann boundary problem for force-free fields

Summary A Neumann boundary value problem for the equation rot –=0 is considered in 29-1 and 29-2. The approach is by transforming the boundary value problem into an equivalent boundary integral equation deduced from a representation formula for solutions of rot –=0 based on the fundamental solution of the Helmholtz equation. In particular, for the two-dimensional case a detailed discussion of the integral equation is carried out including the approximate solution by numerical integration.     

A mixed two-dimensional stationary heat-conduction problem for a cylinder

We investigate the exact solution to the problem of calculating the stationary thermal field external to a cylinder on a portion of the surface of which the thermal flow density is constant and on the remaining portion of which the temperature is constant.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 21, No. 3, pp. 476–479, September, 1971.     

Application of a differential-difference method to the solution of a one-dimensional nonstationary two-layer heat conduction problem with a moving boundary

We apply a differential-difference method to obtain an approximate solution of a nonstationary heat conduction problem with a moving boundary for a medium consisting of an unbounded plate (0xl) and a halfspace (l, 相似文献   

Virtual boundary meshless least square integral method with moving least squares approximation for 2D elastic problem

Moving least squares approximation (MLSA) has been widely used in the meshless method. The singularity should appear in some special arrangements of nodes, such as the data nodes lie along straight lines and the distances between several nodes and calculation point are almost equal. The local weighted orthogonal basis functions (LWOBF) obtained by the orthogonalization of Gramm–Schmidt are employed to take the place of the general polynomial basis functions in MLSA. In this paper, MLSA with LWOBF is introduced into the virtual boundary meshless least square integral method to construct the shape function of the virtual source functions. The calculation format of virtual boundary meshless least square integral method with MLSA is deduced. The Gauss integration is adopted both on the virtual and real boundary elements. Some numerical examples are calculated by the proposed method. The non-singularity of MLSA with LWOBF is verified. The number of nodes constructing the shape function can be less than the number of LWOBF and the accuracy of numerical result varies little.     

Adaptive error estimation technique of the Trefftz method for solving the over-specified boundary value problem

In this paper, numerical solutions are investigated based on the Trefftz method for an over-specified boundary value problem contaminated with artificial noise. The main difficulty of the inverse problem is that divergent results occur when the boundary condition on over-specified boundary is contaminated by artificial random errors. The mechanism of the unreasonable result stems from its ill-posed influence matrix. The accompanied ill-posed problem is remedied by using the Tikhonov regularization technique and the linear regularization method, respectively. This remedy will regularize the influence matrix. The optimal parameter λ of the Tikhonov technique and the linear regularization method can be determined by adopting the adaptive error estimation technique. From this study, convergent numerical solutions of the Trefftz method adopting the optimal parameter can be obtained. To show the accuracy of the numerical solutions, we take the examples as numerical examination. The numerical examination verifies the validity of the adaptive error estimation technique. The comparison of the Tikhonov regularization technique and the linear regularization method was also discussed in the examples.