## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

In the present work, a polygonal boundary element method (PBEM) for solving transient inhomogeneous heat conduction problems with spatially-varying heat generation is developed for the first time. A new general analytical method is proposed and employed in the present PBEM, by which the radial integrals associated with arbitrary spatially-varying density, specific heat, and thermal conductivity can be analytically computed, and the efficiency of the PBEM for solving the transient inhomogeneous problems can be significantly improved. The transient term is dealt with by a finite difference scheme. Three examples are designed to investigate the performance of the proposed method, and the results show that the present method could accurately and efficiently solve transient heat conduction problems with arbitrarily-varying thermal physical properties.

To read the full-text of this research,

you can request a copy directly from the authors.

ResearchGate has not been able to resolve any citations for this publication.

This paper presents a computationally efficient homogenization method for transient heat conduction problems. The notion of relaxed separation of scales is introduced and the homogenization framework is derived. Under the assumptions of linearity and relaxed separation of scales, the microscopic solution is decomposed into a steady-state and a transient part. Static condensation is performed to obtain the global basis for the steady-state response and an eigenvalue problem is solved to obtain a global basis for the transient response. The macroscopic quantities are then extracted by averaging and expressed in terms of the coefficients of the reduced basis. Proof-of-principle simulations are conducted with materials exhibiting high contrast material properties. The proposed homogenization method is compared with the conventional steady-state homogenization and transient computational homogenization methods. Within its applicability limits, the proposed homogenization method is able to accurately capture the microscopic thermal inertial effects with significant computational efficiency.

Recently, a polygonal boundary element method (PBEM) has been developed for solving heat conduction problems. In the PBEM, the radial integration method (RIM) is employed to shift the domain and surface integrals in the boundary-domain integral equation into equivalent line integrals, and all the radial integrals are computed by Gaussian quadrature, which may obtain an accurate solution. However, the computation costs much. Therefore, analytical expressions of radial integrals are derived in this paper, concerning four kinds of varying thermal conductivities and two kinds of heat generation functions, aiming at improving the efficiency. Then all the radial integrals in the boundary-domain integral equation can be analytically calculated. Three examples are employed to examine the analytically-integrated PBEM for solving steady heat conduction problems. The results show that the proposed method is accurate, and it is more efficient than traditional PBEM.

In simulation of heat conduction with temperature-independent physical properties and boundary conditions (BCs), Galerkin residual analysis and variational analysis yield equivalent finite element method (FEM), the conventional FEM. However, if the properties and BCs are temperature-dependent, it is discovered that their derivatives further induce nonlinearity of FEM which consequently generates divergence between the two analyses. A general FEM, extension of variational analysis, is derived as general form of conventional FEM modeling nonlinear heat conduction. Numerical examples demonstrate that the general FEM produces results with considerably higher accuracy and stability and also possesses higher performances on conforming with both two analyses. Since general FEM degenerates to conventional FEM if derivatives are of small-amplitude or zero and its direct implementation to the entire domain is costive, general FEM is alternatively utilized as local refinement of governing equation only to points with significant derivatives. The strategy of local refinement is optimized to enhance efficiency.

In this paper, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed to solve three-dimensional transient heat conduction problems. It is well known that the error of traditional BEM mainly comes from element dispersion, and the introduction of isogeometric ideas makes BEM become a veritable high-precision numerical method. At present, most of the problems solved by isogeometric BEM (IGBEM) are time-independent. The reason is similar to the traditional BEM, which cannot avoid solving domain integrals when solving time-dependent problems. In this paper, based on the potential fundamental solution the boundary-domain integral equation is obtained by the weighted residual method, where the classic dual reciprocity method is adopted to transform domain integrals into boundary integrals. Meanwhile, a two-level time integration scheme is used to solve the discretized differential equations. In addition, the adaptive integration scheme, the radial integral transform method and the power series expansion method are adopted to solve the boundary regular, nearly singular and singular integrals. Several classical numerical examples show that the presented method has good numerical stability and high precision by considering different factors such as the approximation function, the time step, the number of interior points and so on.

Estimating temperature is essential both to design reliable CubeSats and to keep it under maximum operating efficiency. This paper presents the transient thermal simulation of a CubeSat 1U, where the heat transfer by conduction and radiation (external and internal) are solved considering a typical CubeSat mission launched from the International Space Station. The objective is to obtain the temperature field based on the Finite Volume Method (FVM), with inner heat transfer by radiation. The Gebhart method computes the internal heat exchange through successive reflections, and an obstruction model supports the calculation of view factors. Three boundary conditions for the inner side of the CubeSat are tested: without internal heat transfer by radiation, or zero emissivity (ϵ=0.0), with intermediate internal heat transfer by radiation (ϵ=0.5), and maximum internal heat transfer by radiation (ϵ=1.0). The results show a significant impact of the internal heat transfer by radiation in the temperature field of both inner and outer parts of the satellite, and therefore it should not be ignored. Good agreement of transient temperature is found between the FVM and another more straightforward formulation based on the Lumped Method, although the three-dimensional effects are significant and cannot be obtained with such a simplified model.

A new hybrid numerical method for the solution of nonlinear magnetostatic problems in accelerator magnets is proposed. The methodology combines the Fragile Points Method (FPM) and the Boundary Element Method (BEM). The FPM is a Galerkin-type meshless method employing for field approximation simple, local, and discontinuous point – based interpolation functions. Because of the discontinuity of these functions, the FPM can be considered as a meshless discontinuous Galerkin formulation where numerical flux corrections are employed for the treatment of local discontinuity inconsistencies. The FPM possesses the advantages of a mesh-free method, evaluates with high accuracy field gradients, and treats nonlinear magnetostatic problems by utilizing, for the same problem, fewer nodal points than the Finite Element Method (FEM). In the present work, the nonlinear ferromagnetic material of an accelerator magnet is treated by the FPM, while the BEM is employed for the infinitely extended, surrounding air space. The proposed hybrid scheme is based on the scalar potential formulation of Mayergouz et.al (1987), also used in the BEM/BEM and FEM/BEM schemes of Rodopoulos et al. (2019, 2020). The applicability of the method is demonstrated with the solution of representative 2-D magnetostatic problems and the obtained numerical results are compared to those provided by the FEM/BEM scheme of Rodopoulos et al. (2020), as well as by the commercial FEM package ANSYS. Finally, the magnetic field utilized for stable bending of the particles’ trajectory in a 16 Tesla dipole magnet design for the Future Circular Collider (FCC) project of CERN is accurately evaluated with the aid of the proposed here FPM/BEM scheme.

This study presents a coupling formulation for the mechanical modelling of nonhomogeneous reinforced 3D structural systems. The coupling of dissimilar reinforced materials and components provides efficient designs. Particularly, it enables high stiffness and low weight mechanical components, which are largely desired in several engineering applications. In the present study, the Boundary Element Method (BEM) mechanically represents 3D solids through elastic and viscoelastic approaches. The 1D truss BEM elements represent the reinforcements, which allow for elastoplastic behaviour. The sub-region BEM technique models the nonhomogeneous systems whereas connection 1DBEM elements allow for crossing reinforcements along materials interfaces. Bond-slip behaviour has been accounted in the formulation by a nonlinear scheme without special link elements. Different bond-slip laws can be accounted, which enables the mechanical modelling of different adherence behaviours. The proposed formulations are applied in the mechanical analysis of four complex 3D applications. The results achieved by the proposed formulations are compared against the responses of equivalent numerical methods and experimental results available in the literature. The proposed formulations lead to accurate and stable results. Besides, the applications complexity demonstrates it.

A new discrete element-embedded finite element method (DEFEM) scheme is proposed here, which solves the contact force and heat conduction of particles with embedded discrete elements (EDE), employs the finite element method (FEM) to get the deformation and internal temperature change of particles with heat and stress on the boundary and employs the discrete element method (DEM) to obtain the movement of particles. The DEFEM is characterized by coupling the deformation, motion and heat conduction of particles .Compared to either merely DEM or pure FEM, DEFEM combines the solution ideas of the FEM and DEM, and avoids the problem of overlapping and penetration of mesh elements in FEM. DEFEM also supports parallel computing, which is about three times faster than a pure FEM solution. As a demonstration, we developed an in-house code to perform DEFEM to simulate the extrusion and heat conduction of packed pebble bed, in comparison with a pure FEM solution for reference. Based on the numerical results, the characteristics of particle deformation and heat transfer in different extrusion speeds and layers are discussed.

A new approach, radial integration polygonal boundary element method (RIPBEM), for solving heat conduction problems is presented in this paper. The proposed RIPBEM is a new concept in boundary element method (BEM), which would be of great flexibility in mesh generation of complex 3D geometries. Due to the characteristic of arbitrary shapes of polygonal elements, conventional shape functions are insufficient. Moreover, the resulted surface boundary integrals cannot be directly evaluated by the standard Gauss quadrature. To solve these problems, general shape functions for polygonal elements with arbitrary number of nodes are given. To generally and numerically calculate the resulted surface integrals, the radial integration method (RIM) is employed to convert the surface boundary integrals into equivalent contour line integrals of the polygonal elements. As for 3D domain integrals, they are transformed to equivalent line integrals using RIM twice. This methodology can explicitly eliminate strong singularities. Several numerical examples are given to show the effectiveness and the accuracy of the proposed polygonal boundary element method for solving heat conduction problems.

In this paper, a meshless BEM based on the radial integration method is developed to solve transient non-homogeneous convection-diffusion problem with spatially variable velocity and time-dependent source term. The Green function served as the fundamental solution is adopted to derive the boundary domain integral equation about the normalized field quantity. The two-point backward finite difference technique is utilized to discretize the time-dependent terms in the integral equation, which results in that the final integral equation formulation is only related with the normalized field quantity at the current time and has three domain integrals. Both two domain integrals regarding the normalized field quantity at the current and previous times are transformed into boundary integrals by using radial integration method and radial basis function approximation. For domain integral about the source term being known function of time and coordinate, radial basis functions approximation is still adopted to make the transformed boundary integral be evaluated only once, not at each time level. A pure boundary element algorithm with boundary-only discretization and internal points is established and the system of equations is assembled like the corresponding steady problem. Four numerical examples are given to demonstrate the accuracy and effectiveness of the present method.

Natural, forced and mixed convection heat transfer problems are solved by the meshless Method of Approximate Particular Solutions (MAPS). Particular solutions of Poisson and Stokes equations are employed to approximate temperature and velocity, respectively. The latter is used to obtained a closed expression for pressure particular solution. In both cases, the source terms are multiquadric radial basis functions which allow to obtain analytical expressions for these auxiliary problems. In order to couple momentum and energy equations, a relaxation strategy is implemented to avoid convergence problems due to the difference between successive temperature and velocity changes when solving the steady problem from an initial guess. The developed and validated numerical scheme is used to study flow and heat transfer in two two-dimensional problems: natural convection in concentric annulus between a square and a circular cylinder and non-isothermal flow past a staggered tube bundle. Numerical solutions obtained by MAPS are comparable in accuracy to solutions reported by authors who uses denser nodal distributions, showing the capability of the present method to accurately solve heat convection problems with temperature and heat flux boundary conditions as well as curve geometries and internal flow situations.

An integrated geometric design sensitivity (DSA) method for weakly coupled thermoelastic problems is developed in this study using boundary integral equations with an isogeometric approach that directly utilizes a CAD system's NURBS basis functions in response analysis. Thermomechanical coupling frequently creates thermoelastic behaviors in plants and nuclear systems and requires a structural optimization process that minimizes the overall weight and maximizes the system performance. To incorporate accurate geometries and higher continuities into the optimization process, we derive a shape design sensitivity equation using thermoelastic boundary integral equations within the isogeometric framework. In the boundary integral formulation, the shape design velocity field is decomposed into normal and tangential components, which significantly affects the accuracy of shape design sensitivity. Consequently, the developed isogeometric shape DSA method using thermoelastic boundary integral equations is more accurate compared with the analytic solution and the conventional DSA method. Utilizing the formulated isogeometric shape sensitivity as the gradient information of the objective function, isogeometric shape optimization examples for thermoelastic problems are presented. It is demonstrated that the derived isogeometric shape DSA using boundary integral equations are efficient and applicable.

In this paper, a new single interface integral equation method is established for solving non-linear multi-medium heat transfer problems with temperature- dependent thermal conductivity. At first, the boundary-domain integral equation for nonlinear heat transfer in single medium is established based on the fundamental solution of Laplace equation. Then, based on the variation feature of the material properties, a new single integral equation for material nonlinear multi-medium is established according to the degeneration rule from a domain integral to an interface integral. Next, the final system equations are solved by Newton-Raphson iterative method after the discretization. Comparing with conventional multi-domain boundary element method, the presented method is more efficient in computational time, data preparing and program coding. Three numerical examples are given to demonstrate the correctness and robustness of the method presented in the paper.

In this paper, a new framework for stress analysis of three-dimensional (3D) composite (multi-layered) elastic materials is presented. In our computations, the composite material is firstly decomposed into several sub-domains by using the domain-decomposition technique, and then in each of the sub-domain, the stresses are approximated by using a meshless generalized finite difference method (GFDM). Along the sub-domain interfaces, compatibility of displacements and equilibrium of tractions are imposed. The new method yields a sparse and banded matrix system which makes it very attractive for large-scale engineering simulations. Numerical examples with up to 500,000 unknowns are solved successfully using the developed GFDM code.

This paper presents the meshless generalized finite difference method (GFDM) in conjunction with the truncated treatments of infinite domain for simulating water wave interactions with multiple-bottom-seated-cylinder-array structures. In the proposed scheme, the truncated treatments are introduced to deal with the infinite domain. Based on the moving least squares theory and second-order Taylor series expansion, the GFDM approximation formulation is constructed for water wave-structure interactions. It introduces the stencil selection algorithms to choose the stencil support of a certain node from the whole discretization nodes. In comparison with traditional finite difference methods, the proposed GFDM is free of mesh and available for irregular discretization nodes. Numerical investigations are presented to demonstrate the effectiveness of the proposed GFDM with two truncated treatments, absorbing boundary conditions (ABC) and perfectly matched layer (PML), for simulating water wave interactions with single- and four-cylinder-array structures. Then it successfully revisits the near trapped mode phenomenon of specific four-cylinder-array structures and the peak normalized horizontal forces around the cylinders with specific incident water wave. Finally numerical demonstration shows that the structures with porous outer wall can eliminate the near trapped mode phenomenon and reduce the peak normalized horizontal force of multiple-cylinder-array structures.

The paper solves the higher-dimensional inverse heat source problems of nonlinear convection diffusion-reaction equations in 2D rectangles and 3D cuboids, of which the final time data and the Neumann boundary data on one-side are over-specified. Firstly, we derive a family of single-parameter space-time homogenization functions. Secondly, the temperature is obtained through the superposition of the homogenization functions and solving a linear system to satisfy the over-specified Neumann boundary condition. Thirdly, the unknown heat source is recovered by the back substitution of the numerical solution of temperature into the nonlinear governing equation. Numerical tests reveal that the novel method is very accurate to find the solution and to recover the unknown space-time dependent heat source function in the whole space-time domain, whose required extra data are very saving.

It is widely known that the boundary element method (BEM) without any special treatment suffers from the fictitious eigenfrequency problem for the numerical solutions of exterior acoustic problems. This problem has drawn much attention and been extensively studied over the last several decades. However, this paper is concerned with the existence and influence of the fictitious eigenfrequencies when using the BEM for the numerical solutions of interior acoustic problems. To this end, an eigenvalue analysis technique is developed for the acoustic BEM. The nonlinear eigenvalue problem caused by the acoustic BEM is converted into an ordinary linear one by using a contour integral method. Therefore, the conversion is fulfilled by solving a series of BEM systems of equations without any special or complicated treatment of the governing equations or the linear systems. Three interior acoustic examples including two with simply connected domains and one with a multiply connected domain are used to reveal the existence and influence of the fictitious eigenfrequencies. Furthermore, the Burton–Miller formulation with a variable coupling parameter is found to be able to remove such fictitious eigenfrequencies, and the optimal choice of the coupling parameter is investigated.

In this paper, the element differential method is extended to solve a transient nonlinear heat conduction problem with a heat source and temperature-dependent thermophysical properties for the first time. The transient term is discretized by employing a finite difference scheme. An iterative methodology is developed to deal with the nonlinearity caused by temperature-dependent thermophysical properties. Examples of two-dimensional (2D) and three-dimensional (3D) problems are given to validate the present method for solving multi-dimensional transient nonlinear heat conduction problems. The results show that the present EDM provides a promising way that is effective and with high accuracy for solving multi-dimensional transient nonlinear heat conduction problems.

A new boundary domain integral equation with convective heat transfer boundary is presented to solve variable coefficient heat conduction problems. Green’s function for the Laplace equation is used to derive the basic integral equation with varying heat conductivities, and as a result, domain integrals are included in the derived integral equations. The existing domain integral is converted into an equivalent boundary integral using the radial integration method by expressing the normalized temperature as a series of radial basis functions. This treatment results in a pure boundary element analysis algorithm and requires no internal cells to evaluate the domain integral. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method.

In this work, the smoothed finite element method using four-node quadrilateral elements (SFEM-Q4) is employed to resolve underwater acoustic radiation problems. The SFEM-Q4 can be regarded as a combination of the standard finite element method (FEM) and the gradient smoothing technique (GST) from the meshfree methods. In the SFEM-Q4, only the values of shape functions (not the derivatives) at the quadrature points are needed and the traditional requirement of coordinate transformation procedure is not necessary to implement the numerical integration. Consequently, no additional degrees of freedom are required as compared with the original FEM. In addition, the original “overly-stiff” FEM model for acoustic problems (governed by the Helmholtz equation) is properly softened due to the gradient smoothing operations implemented over the smoothing domains and the present SFEM-Q4 possesses a relatively appropriate stiffness of the continuous system. Therefore, the well-known numerical dispersion error for Helmholtz equation is decreased significantly and very accurate numerical solutions can be obtained by using relatively coarse meshes. In order to truncate the unbounded domains and employ the domain-based numerical method to tackle the acoustic radiation in unbounded domains, the Dirichlet-to-Neumann (DtN) map is used to ensure that there are no spurious reflections from the far field. The numerical results from several numerical examples demonstrate that the present SFEM-Q4 is quite effective to handle acoustic radiation problems and can produce more accurate numerical results than the standard FEM.

A double-layer interpolation method (DLIM) is proposed to improve the performance of the boundary element method (BEM). In the DLIM, the nodes of an element are sorted into two groups: (i) nodes inside the element, called source nodes, and (ii) nodes on the vertices and edges of the element, called virtual nodes. With only source nodes, the element becomes a conventional discontinuous element. Taking into account both source and virtual nodes, the element becomes a standard continuous element. The physical variables are interpolated by continuous elements (first-layer interpolation), while the boundary integral equations are collocated at the source nodes only. We further established additional constraint equations between source and virtual nodes using a moving least-squares (MLS) approximation (second-layer interpolation). Using these constraints, a square coefficient matrix of the overall system of linear equations was finally achieved. The DLIM keeps the main advantages of MLS, such as significantly alleviating the meshing task, while providing much better accuracy than the traditional BEM. The method has been used successfully for solving potential problems in two dimensions. Several numerical examples in comparison with other methods have demonstrated the accuracy and efficiency of our method.

This note presents the study of laminar flow under forced convection in buried co-axial exchanger. No usual boundary condition at the exchanger wall is imposed. The wall temperature as well as the wall heat flux and the Nusselt number will be calculated. A hybrid model consisting of a finite element method at the boundary (BEM) for the heat transfer problem on the boundary and a finite volume method (FVM) to solve the laminar flow inside solves this problem. The development of the BEM method is based on the Green's functions theory (GFT). This conjugate method allows to have fast results and to foresee the thermal behavior of the exchanger. The temperature field, the heat flux density and the Nusselt number are investigated. The results is compared to those obtained using the commercial CFD package Fluent The results can be used to improve the heat transfer rate of exchangers. In addition, they can be of great interest in industrial processes requiring the estimation of the heating time necessary to obtain steady states in other similar cases. (c) 2013 Elsevier Ltd. All rights reserved.

This study describes an adaptive finite element methodology for heat transfer by convection applied to microwave heating of liquids. This is the first attempt to model such type of problems employing the concepts of error estimation and mesh adaptivity. The proposed methodology is generic and can be applied to steady-state, transient, linear and nonlinear problems involving heat transfer by conduction and convection. There was very good agreement between simulation and experimental results.

A general algorithm of the distance transformation type is presented in this paper for the accurate numerical evaluation
of nearly singular boundary integrals encountered in elasticity, which, next to the singular ones, has long been an issue
of major concern in computational mechanics with boundary element methods. The distance transformation is realized by making
use of the distance functions, defined in the local intrinsic coordinate systems, which plays the role of damping-out the
near singularity of integrands resulting from the very small distance between the source and the integration points. By taking
advantage of the divergence-free property of the integrals with the nearly hypersingular kernels in the 3D case, a technique
of geometric conversion over the auxiliary cone surfaces of the boundary element is designed, which is suitable also for the
numerical evaluation of the hypersingular boundary integrals. The effects of the distance transformations are studied and
compared numerically for different orders in the 2D case and in the different local systems in the 3D case using quadratic
boundary elements. It is shown that the proposed algorithm works very well, by using standard Gaussian quadrature formulae,
for both the 2D and 3D elastic problems.

In this paper, a simple and robust method, called the radial integration method, is presented for transforming domain integrals into equivalent boundary integrals. Any two- or three-dimensional domain integral can be evaluated in a unified way without the need to discretize the domain into internal cells. Domain integrals consisting of known functions can be directly and accurately transformed to the boundary, while for domain integrals including unknown variables, the transformation is accomplished by approximating these variables using radial basis functions. In the proposed method, weak singularities involved in the domain integrals are also explicitly transformed to the boundary integrals, so no singularities exist at internal points. Some analytical and numerical examples are presented to verify the validity of this method.

A novel combined space-time algorithm for transient heat conduction problems with heat sources in complex geometry

- Liu