Novel approach to compact modeling for nonlinear thermal conduction problems

A novel model order reduction method for the construction of parametric dynamic compact thermal models of nonlinear heat diffusion problems is proposed. The method extends a previous method, recently proposed by the Authors, for the construction of parametric dynamic compact thermal models of linear heat diffusion problems. The method has been validated through the analysis of a simple die in which high levels of non-linearities and large variations of parameters occur.

The goal of this research is to develop a Dynamic Compact Thermal Model (DCTM) of electronic packages. The general objectives of thermal modeling are to increase the accuracy of electrical analysis by taking the effect of temperature variation into consideration, predicting the reliability of the product, and acquiring the information which is necessary to design the cooling system in order to enhance the performance of the electronic systems. The project is focused on generating the dynamic compact thermal model of electronic packages so that the transient thermal behaviors of the package could be predicted fast and accurately. The approach proposed by DELPHI consortium (a collaborative European project) for static compact thermal model generation is extended in this work to generate the dynamic compact model of a BGA package represented by a RC network or admittance matrix. Two steps performed as the methodology of dynamic compact model generation in this work are: 1- A static compact thermal model of BGA package is generated and validated from the static thermal simulation and 2- A RC network is proposed as the contribution of this work and calculated by optimization as the dynamic compact thermal model of the package using the data of transient simualtion. The size of the proposed RC network then is optimized by eliminating some capacitors from the original RC network and validated by comparing its output to the ouptut of finite element simulation. COMSOL©, a Finite element analysis tool is used for thermal simulation and detailed steady state and transient model generation. The optimization algorithm implemented for both static and dynamic compact model generation is Nelder-Mead multidimensional optimization which is realized by MATLAB© programming. The obtained results from the compact models for both static and dynamic analysis of the BGA package are in agreement with the detailed thermal model results and with the available results in literature.

Fully implicit schemes with second‐order time evolutions have been applied to simulate nonlinear diffusion problems precisely for a long time, but there is seldom theoretical study for either their convergence properties or efficient iterations. Here, a second‐order time evolution fully implicit scheme for two‐dimensional nonlinear divergence diffusion problem is analyzed. The unique existence of its solution is given. Two new methods are provided to prove its convergence, including entire inductive hypothesis reasoning and a two‐step reasoning process. Rigorous analysis shows the scheme is stable; its solution has second‐order convergence in both space and time to the exact solution of the problem. The convergence is applied to analyze a Newton iteration accelerating the computation and show its quadratic convergent speed and second‐order accuracy. The reasoning techniques also adapt to first‐order time accuracy schemes, and can be extended to analyze a wide class of nonlinear schemes for nonlinear problems. Numerical tests highlight the theoretical results and demonstrate the high performance of the algorithms. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 121–140, 2016

We address the problem of generating Dynamic Compact Thermal Models for non-linear and non-homogeneous heat diffusion problems. We propose a novel Model Order Reduction technique that is based on i) a specifically designed discretization that properly models the non-linear behaviour of different materials, and ii) the introduction of an improved Hyper Reduction method that dramatically reduces the problem dimension. Our experiments show that our method performs similarly to full Finite Volume simulation in terms of accuracy, while being much more computationally efficient.

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

PurposeThe purpose of this paper is to develop a dynamic compact thermal model (DCTM) of electronic packages. This model is a necessary tool for rapid thermal analysis of the systems which we exposed to boundary condition variation and/or power switching mode such as mobile systems and battery powered systems.Design/methodology/approachThe methodology of compact model generation used was based on generating the transient dynamic detailed finite element thermal model of a package, designing a resistor/capacitor network topology representative of the dynamic detailed model, calculating the resistors'/capacitors' value by optimization method and validation efforts. The method is demonstrated for a ball grid array (BGA) package, a commonly used modern electronic package.FindingsBased on the obtained results, it can be concluded that the dynamic thermal behavior of a BGA package can be accurately described by a generated dynamic compact model in terms of predicted junction temperature response and heat flux of the desired locations of the package.Originality/valueThis model is capable of calculating the temperatures and heat fluxes at desired locations which can help the designer to perform the thermal analysis much faster and easier with the required accuracy.

Based on the generalized functions theory an exact analytical solution to the stationary non-linear problem of thermal conductivity for two-layer plate with heat sources for the asymmetrical Newton’s limiting conditions. Dependence of thermal conductivity factors of each layer on the temperature was assumed to be linear. The problem for a two-layer structure, using the Heaviside asymmetric unit function, is normalized to a single-layer one with piecewise homogeneous (discontinuous) properties of the medium. The advantage of this method for obtaining analytical solutions is that there is no need to directly fulfill the conjugation conditions at the layers’ contact points, since, thanks to the use of the Heaviside function, they are included in the generalized differential equation of the boundary value problem and are fulfilled in the process of obtaining its solution. A specific example of calculating the temperature state of a two-layer plate is provided.

A finite volume method based on radial basis functions for two-dimensional nonlinear diffusion equations

Purpose– Of particular interest is the ability of the extended finite element method (XFEM) to capture transient solution and motion of phase boundaries without adaptive remeshing or moving-mesh algorithms for a physically nonlinear phase change problem. The paper aims to discuss this issue.Design/methodology/approach– The XFEM is applied to solve nonlinear transient problems with a phase change. Thermal conductivity and volumetric heat capacity are assumed to be dependent on temperature. The nonlinearities in the governing equations make it necessary to employ an effective iterative approach to solve the problem. The Newton-Raphson method is used and the incremental discrete XFEM equations are derived.Findings– The robustness and utility of the method are demonstrated on several one-dimensional benchmark problems.Originality/value– The novel procedure based on the XFEM is developed to solve physically nonlinear phase change problems.

A qualitative analysis and numerical simulations of a nonlinear second-order anisotropic diffusion problem with non-homogeneous Cauchy–Neumann boundary conditions

This article employs the continuous-time analog Hopfield neural network (CHNN) to compute the temperature distribution in nonlinear heat conduction problems. The relationship between the CHNN synaptic connection weights and the governing equations of the nonlinear heat conduction problems is established and a corresponding network connectivity circuit design scheme proposed. The CHNN algorithm is used to solve the heat equation for conduction problems with a power-law nonlinearity. The results confirm that the proposed CHNN scheme provides an accurate means of solving the transient temperature distributions of nonlinear heat conduction problems on a real-time basis.

Hybrid Laplace transform technique for non-linear transient thermal problems

Estimation of concentration-dependent diffusion coefficient in pressure-decay experiment of heavy oils and bitumen

A novel approach for extracting dynamic compact thermal models from nonlinear heat diffusion equations, which take into account the temperature dependence of thermal properties, is presented. In such Model Order Reduction method, the nonlinear heat diffusion equation is projected onto the space spanned by a few terms in the Volterra's series expansion of the solution. Such projection is performed by a novel structure-preserving hyper-reduction algorithm. The resulting method is very efficient, and can lead to very accurate compact thermal models, with unprecedented levels of simplicity. Moreover such models can be extracted for practically any regular temperature dependence of heat capacity, thermal conductivity and heat exchange coefficients. The method has been validated through the analysis of a complex ultra thin stacked chip module.

Boiling in cooling channels has recently been demonstrated to be an effective mechanism for heat extraction in pressure die casting. Boiling heat transfer can be enhanced by cooling channel shape optimization. The occurrence of boiling presents a non-linear thermal problem which, when combined with shape optimization, necessitates the solving of non-linear equations for each channel configuration. In this paper a methodology is presented that involves the use of optimization for the combined determination of channel shapes and heat transfer coefficients. It is shown in the paper how this approach results in the accurate determination of boiling heat transfer coefficients on the final optimized cooling channel configuration. The non-linear thermal problem is calculated at very little computational cost over that required for a comparable linear problem. Focus in the paper is on the application of the methodology to the pressure die casting process. The approach adopted is founded on a design sensitivity analysis using the material derivative adjoint variable method. The thermal model for the pressure die casting process is founded on the boundary element method and the optimization is performed using a conjugate gradient scheme. Geometrical constraints are enforced using buffer elements superimposed on to the boundary element mesh. Numerical and experimental trials are performed to demonstrate the potential of the new optimization methodology.