Ryszard A. Białecki

Solving inverse heat conduction problems using trained POD-RBF network inverse method

The article presents advances in the approach aiming to solve inverse problems of steady state and transient heat conduction. The regularization of ill-posed problem comes from the proper orthogonal decomposition (POD). The idea is to expand the direct problem solution into a sequence of orthonormal basis vectors, describing the most significant features of spatial and time variation of the temperature field. Due to the optimality of proposed expansion, the majority of the basis vectors can be discarded practically without accuracy loss. The amplitudes of this low-order expansion are expressed as a linear combination of radial basis functions (RBF) depending on both retrieved parameters and time. This approximation, further referred as trained POD-RBF network is then used to retrieve the sought-for parameters. This is done by resorting to least square fit of the network and measurements. Numerical examples show the robustness and numerical stability of the scheme.

Solving Transient Nonlinear Heat Conduction Problems by Proper Orthogonal Decomposition and the Finite-Element Method

ABSTRACT A method of reducing the number of degrees of freedom and the overall computing time by combining proper orthogonal decomposition (POD) with the finite-element method (FEM) has been devised. The POD-FEM technique can be applied both to linear and nonlinear problems. At the first stage of the method a standard FEM time-stepping procedure is invoked. The temperature fields obtained for the first few time steps undergo statistical analysis, yielding an optimal set of globally defined trial and weighting functions for the Galerkin solution of the problem at hand. The resulting set of ordinary differential equations (ODEs) is of greatly reduced dimensionality when compared with the original FEM formulation. For linear problems, the set can be solved either analytically, resorting to the modal analysis technique, or by time stepping. In the case of nonlinear problems, only time stepping can be applied. The focus of this article is on the time-stepping approach, in which the generation of the FEM-POD matrices, requiring some additional matrix manipulations, can be embedded in the assembly of standard FEM matrices. The gain in execution times comes from the significantly shorter time of solution of the set of algebraic equations at each time step. Numerical results are presented for both linear and nonlinear problems. In the case of linear problems, the derived time-stepping technique is compared with the standard FEM and the modal analysis. For nonlinear problems the proposed POD-FEM approach is compared with the standard FEM. Good accuracy of the POD-FEM solver has been observed. Controlling the error introduced by the reduction of the degrees of freedom in POD is also discussed.

Heat transfer analysis of the continuous casting process by the front tracking BEM

Abstract Heat transfer problem in the strand during continuous casting is considered in the paper. Steady state temperature field in a coordinate system attached to the mould has been analyzed. Phase change is assumed to take place at constant temperature. The nonlinearity in this problem is caused by the unknown location of the solid–liquid interface. To determine both, this location and the resulting temperature field in the strand, a novel BEM front tracking algorithm has been developed. Numerical examples like one-dimensional (1D) phase change benchmark (modeled as two-dimensional (2D)) and several 2D problems of continuous casting of copper are included and discussed.

Dual reciprocity BEM without matrix inversion for transient heat conduction

Abstract Presence of domain integrals in the formulation of the boundary element method dramatically decreases the efficiency of this technique. Dual reciprocity boundary element method (DRBEM) is one of the most popular methods to convert domain integrals into a series of boundary integrals. This is done at the expense of generating some additional matrices and inverting one of them. The latter feature makes the DRBEM inefficient for large-scale problems. This paper describes simple means of avoiding matrix inversion for transient heat transfer problems with arbitrary set of boundary conditions. The technique is also directly applicable to other phenomena (acoustic wave propagation, elastodynamics). For the boundary conditions of Neumann and Robin type, the proposed technique produces exactly the same results as the standard approach. In the presence of Dirichlet conditions, a lower bound on the time step has been detected in the backward difference time stepping procedure. The approach has been tested on some transient heat conduction benchmark problems and accurate results have been obtained.

Iterative solution of large-scale 3D-BEM industrial problems

In the present study we demonstrate that a standard direct boundary integral formulation and the collocation method result, for a wide range of practical applications, in systems of equations having properties that allow efficient use of Krylov iterative solvers with simple diagonal and block-diagonal preconditioning. In practical problems the matrix diagonal dominance, especially in its strong sense, is never observed when unstructured meshes and complex geometry are used. However, from our investigations we obtained that the diagonal dominance is not so important and matrix properties desired for a good convergence are rather associated with eigenvalues distribution. A simple diagonal scaling of a properly structured BEM matrix can improve this distribution dramatically. For industrial BEM packages the data sets are generated via preprocessors. For iterative solvers, in contrast to direct solvers, it is very important to prepare a system of equation in an appropriate form. We show that a correct treatment of boundary conditions and proper ordering of unknowns and equations are essential to obtain convergence. Extensive numerical experiments with preconditioned GMRES(m) and CGS methods for large practical problems have been carried out.

Temperature in a disk brake, simulation and experimental verification

Purpose – This paper seeks to develop a reliable simulation technique and experimental equipment applicable to thermal analysis of disk brakes. The application is focused on safety issues arising in coal mines and other hazardous explosive environments.Design/methodology/approach – The experimental rig provides data on the friction power generated by the disk‐pad pair for a user‐defined squeezing force program. The developed software predicts the temperature field in the brake and pad. The code is based on the finite volume approach and is formulated in Lagrangian coordinates frame.Findings – In the circumferential direction advection due to the rotation of the disk dominates over the conduction. The energy transfer problem could be formulated in a Lagrange coordinates system as 2D. A novel approach to the estimation of the uncertainty of numerical simulations has been proposed. The technique is based on the GUM methodology and uses sensitivity coefficients determined numerically. Very good agreement of s...

CFD Two-Scale Model of a Wet Natural Draft Cooling Tower

A 2-D axisymmetric model of multiphase heat, mass, and momentum transfer phenomena in natural draft cooling tower is developed using a CFD code Fluent. The fill of the tower is modeled as a porous medium. The energy and mass sources in this zone are evaluated solving a separate 1-D model of mass and heat exchange. The spatial dependence of the sources is accounted for by dividing the fill into a set of vertical channels. The CFD solver produces boundary conditions for each channel, while the model of the channel exports the heat and mass sources to the CFD solver. To accelerate the calculations, an original technique known as the proper orthogonal decomposition (POD) is applied. This approach produces a reduced dimensionality model resulting in significant time economy and accuracy loss lower than 2%. The Euler-Euler multiphase model is used in the rain zone. The simulation results have been validated against experimental data coming from field measurements of a large industrial installation.

SOLVING NONLINEAR STEADY-STATE POTENTIAL PROBLEMS IN INHOMOGENOUS BODIES USING THE BOUNDARY-ELEMENT METHOD

A n efficient algorithm for solving nonlinear steady-state potential problems in multilayered media using the boundary-element method (BEM) is presented. Two sources of nonlinearity are considered: solution-dependent material properties and nonlinear boundary conditions, The approach is based on Kirchhoffs transformation. Applying this transformation and a standard BEM technique results in a set of nonlinear equations. The final version of the developed nonlinear equations solver evolved from some simple iteration schemes that proved to be divergent or slowly convergent. The solver relies on the incremental loading concept and Aitken extrapolation. Numerical examples dealing with thermal analysis of industrial furnaces are included. A simple and efficient Kirchhoffs transformation and inverse transformation technique is described.

Minimum distance calculation between a source point and a boundary element

Abstract The accuracy of integration over boundary elements determines the accuracy of the final results. Contemporary boundary element method (BEM) codes use adaptive integration to limit the numerical effort of forming BEM matrices. To accomplish this the minimum distance between the source point and the element is needed. The paper presents an efficient and reliable algorithm for determining this distance. Numerical examples showing the robustness of the algorithm and the applied adaptive integration scheme are included.

Application of the Proper Orthogonal Decomposition in Steady State Inverse Problems

A novel inverse analysis technique for retrieving unknown boundary conditions has been developed. The first step of the approach is to solve a sequence of forward problems made unique by defining the missing boundary condition as a function of some unknown parameters. Taking several combinations of values of these parameters produces a sequence of solutions (snapshots) which are then sampled at a predefined set of points. Proper Orthogonal Decomposition (POD) is used to produce a truncated sequence of orthogonal basis functions, being appropriately chosen linear combinations of the snapshots. The solution of the forward problem is then written as a linear combination of the basis vectors. The unknown coefficients of this combination are evaluated by minimizing the discrepancy between the measurements and the POD approximation of the field. Two numerical examples show the robustness and numerical stability of the proposed scheme.