Radial Integration Boundary Element Method Research Articles

POD method for solving elastodynamics problems of functionally gradient materials based on radial integration boundary element method

Functionally graded materials (FGMs) are widely utilized due to their excellent mechanical properties, prompting a need to explore the dynamic response of FGMs. To accurately and efficiently analyze elastodynamics problems, the radial integration boundary element method (RIBEM) is coupled with the proper orthogonal decomposition (POD) method in this paper. The POD method can transform a high-dimensional model into a low-dimensional system with high precision. Using the basic solution (Kelvin solution), the governing partial differential equation for the linear elastic dynamics problem of FGMs is transformed into a boundary-domain integral equation. The radial integration method is employed to convert the domain integral arising from material inhomogeneity and inertia terms into an equivalent boundary integral. This establishes a solution for the elastic dynamics problem of FGMs without the need for internal grid RIBEM. The Newmark time integration scheme is applied to solve the second-order ordinary differential equations. The displacement field obtained by RIBEM is utilized to construct the snapshots matrix, from which the POD reduced-order model is established. Numerical examples validate that the POD method significantly reduces the model's order, enhancing computational efficiency while also demonstrating the high accuracy of the POD method.

A Fast Reduced-Order Model for Radial Integration Boundary Element Method Based on Proper Orthogonal Decomposition in the Non-Uniform Coupled Thermoelastic Problems

To efficiently address the challenge of thermoelastic coupling in functionally graded materials, we propose an approach that combines the radial integral boundary element method (RIBEM) with proper orthogonal decomposition (POD). This integration establishes a swift reduced-order model to transform the high-dimensional system of equations into a more manageable, low-dimensional counterpart. The implementation of this reduced-order model offers the potential for rapid numerical simulations of functionally graded materials (FGMs) under thermal shock loading. Initially, the RIBEM is utilized to resolve the thermal coupling issue within the FGMs. From these solutions, a snapshot matrix is constructed, capturing the solved temperature and displacement fields. Subsequently, the POD modes are established and a POD reduced-order model is constructed for the boundary element format of the thermally coupled problem. Finally, a system of low-order discrete differential equations is solved. Numerical experiments demonstrate that the results obtained from the reduced-order model closely align with those of the full-order model, even when considering variations in structural parameters or impact loads. Thus, the introduction of the reduced-order model not only guarantees solution accuracy but also significantly enhances computational efficiency.

A reduced-order modeling for thermo-mechanical coupling analyses by using radial integration boundary element method

In order to efficiently solve thermo-mechanical coupling problems, this work combines the radial integration boundary element method (RIBEM) and the proper orthogonal decomposition (POD) to construct a fast reduced-order model. This model transforms high-dimensional systems into low-dimensional ones, enabling faster computation of thermo-mechanical coupling problems under thermal loads. Firstly, RIBEM is employed to solve coupled thermoelastic dynamic problems under thermal shock loads. Domain integrals are converted to boundary integrals using the radial integration method, establishing a purely boundary element algorithm without internal cells. Subsequently, finite difference techniques are applied to calculate the discretized algebraic equations, and the obtained field variables are used as a snapshot matrix to establish POD modes and construct the POD reduced-order model. Numerical examples demonstrate that the results of the reduced-order model are essentially consistent with the full-order model across various physical parameters and boundary conditions. Therefore, this reduced-order model significantly improves computational efficiency while maintaining high accuracy, especially when solving large-scale problems.

The solution of elastodynamic problems using modified radial integration boundary element method (MRIBEM)

In this paper, the modified radial integration boundary elements method (MRIBEM) is applied to analyze two-dimensional (2D) elastodynamic problems. The boundary and domain variables are approximated by Lagrangian shape functions and global radial basis functions (RBFs), respectively. As a key idea of this study, the domain integrals that result from implying the inertia force effect are computed by means of a modified radial integration method (MRIM). With the MRIBEM, we are able to deal with problems with concave shapes when the domain variables are approximated by global RBFs by defining an auxiliary point or, in case of complex domain shapes, two or more auxiliary points as the origin for radial integration. Two-time marching schemes, i.e. Newmark method and the Houbolt method, are implemented for the solution of governing differential equations along time. The performance of the proposed method is examined in the solution of four numerical examples.

Numerical solution of water wave propagation problems over variable bathymetries using the modified radial integration boundary element method

In this paper, a modified radial integration boundary element method (MRIBEM) is applied to the solution of water wave propagation problem. The mild slope equation which has proved to be a very efficient equation for a large number of water wave problems is solved using the MRIBEM over arbitrary seabed geometries. The infinite region is subdivided into a central finite area with irregular seabed and a semi-finite one with parallel depth contour lines. The interior part is dealt with using MRIBEM and the exterior one is solved using the boundary element method. This method benefits from the meshfree characteristic of MRIBEM for the inner domain and the fulfillment of Sommerfeld's radiation condition for the exterior domain at the same time. Numerical examples are compared with experimental results and other researches which are indicative of the validity and efficiency of the present method.

Shape reconstruction in transient heat conduction problems based on radial integration boundary element method

In order to accurately identify the geometric boundary, the radial integration boundary element method (RIBEM) combined with the modified Levenberg-Marquardt (LM) algorithm is proposed for shape reconstruction in transient heat conduction problems. Compared with the finite element method (FEM), the boundary element method (BEM) only discretizes the boundary rather than the whole domain, so it has incomparable advantages in the shape reconstruction. Especially, the radial integration method still maintains the superiority of boundary discretization in transient heat conduction problems. For the iterative scheme of LM algorithm, the shape reconstruction greatly depends on whether the sensitivity matrix can be obtained accurately. Therefore, complex variable derivation method (CVDM) is firstly introduced to carry out shape reconstruction in the transient heat conduction problems, which accurately solves the sensitivity coefficients of node temperature to the reconstructed boundary, and the RIBEM is transformed from the real domain to the complex domain to solve the transient heat conduction problems. The modified LM algorithm effectively overcomes the shortcoming of the excessive dependence on the step size of finite difference scheme. It should be noted that compared with the finite difference scheme, the proposed method can greatly reduce the computational cost of the direct problem in the multi-variate optimization process, thereby greatly improving the efficiency of identifying the boundary shape. Finally, different reconstruction strategies are applied in numerical models with three types of variable boundary, and reconstruction results demonstrate the great accuracy and efficiency of the proposed method in 2D and 3D shape reconstruction, and it also shows good robustness in the presence of measurement errors.

Shape Reconstruction in Transient Heat Conduction Problems Based on Radial Integration Boundary Element Method

Shape Reconstruction in Transient Heat Conduction Problems Based on Radial Integration Boundary Element Method

A fast reduced-order model for radial integral boundary element method based on proper orthogonal decomposition in nonlinear transient heat conduction problems

In order to efficiently solve nonlinear transient heat conduction problem, a method combining the radial integration boundary element method (RIBEM) and the proper orthogonal decomposition (POD) is proposed to establish the nonlinear reduced-order model, and the implementation of the reduced-order model makes fast numerical simulation possible in engineering field. Transient heat conduction problems with temperature-dependent and temperature-independent thermal conductivity are firstly solved by the RIBEM respectively, and two types of snapshots are composed of these solved transient temperature fields. Subsequently, the POD is applied in the analysis of the snapshots, and then a set of optimal orthogonal basis can be obtained. In the procedure of establishing the reduced-order model, we need to redistribute the nonlinear equations. The strategy is to separate the boundary nodes defined as Dirichlet condition, and then the orthogonal basis is used to derive a reduced-order expression for the remaining unknowns. Therefore, the degree of freedom that needs to be solved can be greatly reduced. Numerical examples show that different reduced-order models based on nonlinear basis and linear basis are all consistent with the full-order model. What is more, the computational efficiency is greatly improved due to the introduction of the reduced-order model.

Precise time-domain expanding boundary element method for solving phase change problems

In this article, the precise time-domain expanding boundary element method (PTEBEM) is applied to solving phase change problems. In the present method, the recursive formulation of the governing equation is derived by expanding time-dependent physical quantities at discrete time intervals. Then the recursive equation is solved by the radial integration boundary element method, where the radial integral method is used to transform the domain integral into the boundary integral. Moreover, in order to guarantee the accuracy and efficiency, a self-adaptive check scheme is introduced to evaluate the recursive times in a time step. Simultaneously, the front-tracking method is used to simulate the motion of the moving boundary. To accomplish this purpose, an iterative implicit solution algorithm has been developed by employing PTEBEM. To verify the present method, several typical examples are given. Satisfactory results are obtained in comparison with reference solutions.

A new modified Levenberg-Marquardt algorithm for identifying the temperature-dependent conductivity of solids based on the radial integration boundary element method

In this paper, combination of radial integration boundary element method (RIBEM) with complex variable and Levenberg-Marquardt algorithm (LMA) is firstly proposed to identify temperature-dependent conductivity in the inverse heat conduction problem. To obtain the simulative temperature, radial integration boundary element method is used to solve the transient nonlinear heat conduction problem with temperature-dependent conductivity. What’s more, RIBEM with complex variable, which transforms the real variables into complex ones in boundary element method, makes it possible to perform complex variable derivative method (CVDM) in LMA. Furthermore, because of the introduction of CVDM, the sensitivity coefficient matrix can be calculated accurately and efficiently, and then the identification of unknown variable can be achieved admirably in the inverse heat problem. Finally, different initial guess value and measurement errors are considered, respectively, and various numerical examples are presented to fully demonstrate the accuracy and feasibility of the proposed method in identifying temperature-dependent conductivity.

Numerical solution of scalar wave equation by the modified radial integration boundary element method

A radial integration boundary element method is applied to the analysis of two-dimensional scalar wave equation problem. Based on two kinds of fundamental solutions, two types of the integral equation is derived for solution of wave problem. Time domain integrals which are imposed to the solution are implemented using an efficient modified formulation of the radial integration method for concave shapes which was recently introduced in [1]. This boundary only element approach is applied to Newmark and Houbolt time stepping methods. Performance of the implemented formulation is evaluated using several numerical examples.

Radial integration boundary element method for solving two-dimensional unsteady convection–diffusion problem

Numerical solution of two-dimensional unsteady convection–diffusion problem is carried out by using the radial integration boundary element method in the paper. A boundary domain integral equation is established for the purpose by using the fundamental solution of Laplace equation. The convective and time-dependent terms of governing equation lead to the appearance of two domain integrals including the unknown quantities in the integral equation. The radial integration method is used to transform the two domain integrals into the equivalent boundary integrals over the global boundary by approximating the unknown quantities through the augmented fourth-order spline radial basis function. Thus, a pure boundary element algorithm with the requirement of boundary-only discretization and some internal points instead of internal cells is developed. The finite difference scheme for discretizing the time-dependent term is utilized to assemble the final system of equations. Several numerical examples are given to illustrate the accuracy and efficiency of the proposed method.

Radial integration boundary element method for two-dimensional non-homogeneous convection–diffusion–reaction problems with variable source term

This paper describes a new formulation of the radial integration boundary element method (RIBEM) for two-dimensional non-homogeneous convection–diffusion–reaction problems with variable source terms. The radial integration method (RIM) is implemented to transform the resulting domain integral into equivalent boundary integrals, and thus a boundary-only integral equation formulation can be achieved. The fundamental solution of the steady-state convection–diffusion–reaction equation with constant coefficients is employed. The integral representation formula for the convection–diffusion–reaction problem with source term is obtained from Green’s second identity. Numerical applications are included for five different cases, for which analytical solutions are available, to establish the validity of the proposed approach and to demonstrate convergence and efficiency of the proposed technique. Results obtained show that the RIBEM produced an excellent agreement with the analytical solutions and the results do not present oscillations or damping of the wave front, as may appear in other numerical techniques.

A new modified conjugate gradient method to identify thermal conductivity of transient non-homogeneous problems based on radial integration boundary element method

In order to analyze the heat conduction problem of composite materials in aerospace engineering, a modified conjugate gradient method is proposed to identify the physical parameters of transient heat conduction problems in this paper. In the positive problem, the boundary element method based on radial integration method is used to obtain the measured point temperature of the transient heat conduction problem with varying thermal conductivity. In the inverse problem, based on the advantages of high accuracy and fast convergence rate of the conjugate gradient method, a modified conjugate gradient method is used. On the one hand, the complex variable derivative method is introduced into the traditional conjugate gradient method, and then the sensitivity coefficient matrix can be accurately calculated. On the other hand, based on the linear convergence property of the conjugate gradient method, a restart factor is introduced in this paper. For an inverse problem containing n unknown parameters, the algorithm restarts from the steepest descent method after every n + 1 iterations to identify physical properties of transient non-homogeneous thermal conduction. The complex variable derivation method overcomes the problem that the traditional conjugate gradient method needs differential processing and low derivative accuracy in solving the inverse transient heat conduction problem. When solving the complicated inverse problems, conjugate gradient method, which includes restart factor, solves the problem of slow convergence in traditional conjugate gradient methods. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method in identifying the parameters of transient non-homogeneous thermal conductivity.

Dynamic fracture analysis of functionally graded materials under thermal shock loading by using the radial integration boundary element method

A coupled thermoelastic radial integration boundary element method is applied to analyze the dynamic fracture mechanics of functionally graded materials (FGMs) subjected to the thermal shock loadings. The dynamic stress intensity factor (DSIF) of the crack tip is defined by the crack open displacement (COD) near the crack tip. The effects of material gradating direction versus crack direction and the coupling effects on the stress intensity factor are studied for two- and three-dimensional crack structures. The results demonstrate that the present method is very accuracy and efficiency to analyze the dynamic fracture mechanics for the cracked FGMs. The research also provides a theoretical basis for engineering design, and can extend the application range of the boundary element method.

Radial integration BEM for steady convection-conduction problem with spatially variable velocity and thermal conductivity

In this paper, the radial integration boundary element method without internal cells is presented for solving steady convection-conduction problem with spatially variable velocity and thermal conductivity. The temperature boundary integral equation is derived by employing the fundamental solution for the potential problem (Green function) as well as the normalized temperature and thermal conductivity, which results in the appearance of domain integrals including the unknown temperature. To avoid evaluating the domain integrals with the use of internal cells, the transformation of domain integrals to the boundary is carried out by employing the radial integration method and approximating the normalized temperature with the use of the compactly supported fourth-order spline radial basis function combined with the polynomials in global coordinates. Finally, a pure boundary element algorithm with some selected internal nodes is established for the analysis of steady convection-conduction problems with spatially variable velocity and thermal conductivity. Three numerical examples are given to demonstrate the accuracy and effectiveness of the present method.

A radial integration boundary element method for solving transient heat conduction problems with heat sources and variable thermal conductivity

ABSTRACTA new radial integration boundary element method (RIBEM) for solving transient heat conduction problems with heat sources and variable thermal conductivity is presented in this article. The Green’s function for the Laplace equation is served as the fundamental solution to derive the boundary-domain integral equation. The transient terms are first discretized before applying the weighted residual technique that is different from the previous RIBEM for solving a transient heat conduction problem. Due to the strategy for dealing with the transient terms, temperature, rather than transient terms, is approximated by the radial basis function; this leads to similar mathematical formulations as those in RIBEM for steady heat conduction problems. Therefore, the present method is very easy to code and be implemented, and the strategy enables the assembling process of system equations to be very simple. Another advantage of the new RIBEM is that only 1D boundary line integrals are involved in both 2D and 3D problems. To the best of the authors’ knowledge, it is the first time to completely transform domain integrals to boundary line integrals for a 3D problem. Several 2D and 3D numerical examples are provided to show the effectiveness, accuracy, and potential of the present RIBEM.

RIBEM for 2D and 3D nonlinear heat conduction with temperature dependent conductivity

In this paper, a new and effective radial integration boundary element method (RIBEM) is presented to solve nonlinear heat conduction with temperature dependent thermal conductivity of materials. Boundary-domain integral equation is formulated for nonlinear heat conduction by utilizing the fundamental solutions for the corresponding linear heat conduction, which contains a domain-integral due to the temperature dependence of the thermal conductivity of the materials. Two different approaches based on the radial basis functions are implemented to approximate the unknowns appearing in domain integrals. The arising domain-integral is converted into the equivalent boundary integrals using the radial integration method (RIM), resulting in a pure boundary element analysis algorithm. Newton−Raphson iterative method is applied to solve the final system of algebraic equations after the discretization. Numerical examples are presented to demonstrate the accuracy and the efficiency of the present method.

Radial integration boundary element method for heat conduction problems with convective heat transfer boundary

ABSTRACTA 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.

Analytically-integrated radial integration BEM for solving three-dimensional transient heat conduction problems

This paper presents a new strategy using radial integration boundary element method (RIBEM) to solve three-dimensional transient heat conduction problems. In this method, the radial integral, which is used to transform the domain integrals into equivalent boundary integrals, is analytically integrated by using some newly proposed analytical expressions. This analytical approach can improve the computational efficiency and computational accuracy considerably compared with traditional RIBEM in evaluating the domain integrals through using the radial integration method (RIM). The Green's function for the Laplace equation is utilized as the fundamental solution to derive the boundary-domain integral equation for transient heat conduction. RIM is used to convert the domain integrals associated with the time derivative of temperature into equivalent boundary integrals. The derivation process about the analytical radial integral expressions published before is further investigated, the adequate and simple expression for settling the special circumstances in which the newly derived analytical expressions become invalid, is deduced in this paper. Numerical examples are given to demonstrate the efficiency of the presented method.