Boundary-domain Integral Equations 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.

Computation of Neumann Localised Boundary Domain Integral Equations

Most integrals in Localised Boundary Domain Integral Equations (LBDIEs) comprise singularities. This paper aims to produce numerical solutions of the LBDIEs for the Partial Differential Equations with variable coefficients. The singularities of the boundary integrals in LBDIEs will be handled by using a semi-analytic for logarithmic singularity and a semi-quadratic analytic method for r−2 singularity. Whereas the singular domain integrals are handled by using the Duffy transformation. The LBDIEs that we consider are associated with the Neumann problem, which can be solved with a condition. If it can be solved, the solution is, however, unique up to an additive constant. We add a perturbation operator to the LBDIEs to convert the LBDIE to a uniquely solvable equation. The perturbed integral operator leads the perturbed LBDIEs to a dense matrix system that disable the use of methods in solving sparse matrix system. We solve the system of linear equations by Lower-Upper (LU) decomposition method. The numerical results indicate that high accuracy results can be attained. It gives the impression that the methods we use in this numerical experiment are reliable in handling the boundary and domain singular integrals.

Analysis of two-operator boundary-domain integral equations for variable-coefficient Dirichlet and Neumann problems in 2D with general right-hand side

Analysis of two-operator boundary-domain integral equations for variable-coefficient Dirichlet and Neumann problems in 2D with general right-hand side

Isogeometric boundary element method for isotropic damage elastic mechanical problems

In this work, a complex isotropic damage mechanics structural model is developed using the isogeometric method to study the strain displacement and influence factors in isotropically damaged elastics. The proposed modeling method can be constructed dynamically by introducing control points and weight coefficients, which improves the accuracy and eliminates mesh errors, and the boundary element method is used for theoretical analysis and calculation of damage mechanics. For the description of irregular geometric models, the isogeometric method has unparalleled superiority. The isogeometric method can seamlessly connect the geometry generated by the advantages of CAD and CAE, and the combination of the two methods is very useful for the future extension of IGABEM to practical engineering applications. At the same time, we use the source point isolation method to establish the boundary-domain integral equations and use the radial integration method to solve the domain integrals. Compared with the traditional boundary element method, the isogeometric boundary element method makes the construction of complex models simple. The numerical results show that the numerical method has good stability and robustness, which provides strong support for further software integration.

RI-CDI-FDTD method and program implementation for electromagnetic characteristics simulation of lossy Debye dispersive medium

Dispersive media refer to a class of natural substances, including living organisms, composite materials, plasma and water, with diverse applications in areas such as biomedicine, microwave sensing, electromagnetic protection, and stealth technology. In the pursuit of investigating the electromagnetic properties of these media, time-domain numerical methods, including finite difference in time domain (FDTD), finite element method, and time domain boundary integral equation method, have been widely utilized. Time-domain numerical methods are preferred to their frequency-domain counterparts owing to their ability to handle nonlinear and wideband problems, as well as various material properties. The FDTD method, in particular, is a highly adaptable, robust, and easy-to-use numerical method that directly solves the Maxwell equations while also simulating the reflection, transmission, and scattering of electromagnetic waves in complex dispersion media. Nonetheless, the traditional FDTD method suffers low computational efficiency arising from the Courant-Friedrichs-Lewy (CFL) stability condition. To solve the problem of low computational efficiency, a new method, the complying divergence implicit finite-difference time-domain (CDI-FDTD) method with a one-step leapfrog scheme, is introduced for lossy Debye dispersive media. The Maxwell equations in the frequency domain form a starting point, and the Fourier transform is utilized to transform the electromagnetic field components from the frequency domain to the time domain. To approximate the integral terms arising from the frequency-to-time domain transformation, a recursive integration (RI) method is employed. Subsequently, the time-domain Maxwell equations and auxiliary variables are discretized with a one-step leapfrog implicit scheme. The iterative formula of the RI-CDI-FDTD algorithm for lossy Debye dispersive media is then derived. The RI-CDI-FDTD method does not change the formulas of the traditional CDI-FDTD method while only requiring to add auxiliary variables for updating field components to the dispersive medium region. The numerical implementation is straightforward, and the electromagnetic modeling is flexible. Moreover, the unconditional stability of the RI-CDI-FDTD algorithm is proven by using the von Neumann method. Finally, some numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed method. In conclusion, our work contributes a crucial numerical simulation tool to accurately modeling complex dispersive media while providing a systemic stability analysis method for time-domain numerical methods.

Isogeometric dual reciprocity BEM for solving non-Fourier transient heat transfer problems in FGMs with uncertainty analysis

In this paper, the precise integration multi-patch isogeometric dual reciprocity boundary element method (IG-DRBEM) of the non-Fourier transient heat transfer problem in functionally graded materials (FGMs) is proposed, and the stochastic analysis of multidimensional material uncertainty by hyperbolic truncated adaptive sparse polynomial chaos expansion (PCE) is established based on the determine theory. Up to now, most research results of the isogeometric analysis boundary element method (IGABEM) in transient heat transfer problems are based on Fourier's law. The main reason is the higher-order derivative term with respect to time in the non-Fourier heat transfer control equation leads to additional domain integrals in the boundary-domain integral equation. The dual reciprocity method (DRM) can transform the domain integral to the boundary conveniently and maintain the dimension reduction advantage of the BEM. The introduction of the precise integration method (PIM) can improve the stability and accuracy of the time-domain solution. In the random analysis based on IG-DRBEM, a sparse PCE strategy with hyperbolic truncation is constructed to overcome the dimensional disaster difficulty of traditional PCE. Through the proposed adaptive iterative process, a relatively small number of PCE terms are retained compared with the full PCE on the premise of meeting the error estimation requirements. The presented method is verified to have good accuracy and high efficiency by the complex model similar to actual engineering, which further expands the application extent of IG-DRBEM.

A time domain boundary element method for acoustic scattering by lined surfaces in a subsonic uniform mean flow

A time domain boundary integral equation with Burton-Miller reformulation is presented for acoustic scattering by surfaces with liners in a uniform mean flow. The Ingard-Myers impedance boundary condition is implemented using a broadband multipole impedance model which is in turn converted into time domain differential equations to augment the boundary integral equation. The coupled integral-differential equations are solved numerically by a March-On-in-Time (MOT) scheme. While the Ingard-Myers condition is known to support Kelvin-Helmholtz instability due to its use of a vortex sheet interface between the flow and the lined surface, it is found that by neglecting a second-order derivative term in the current time domain impedance boundary condition formulation, the instability can be effectively avoided in computation. Proposed formulation and implementation are validated with numerical examples. Moreover, a minimization procedure for finding the poles and coefficients of the broadband multiple impedance expansion is formulated by which, unlike the commonly used vector-fitting method, passivity of the model is ensured. Numerical tests show the proposed minimization approach is effective for modeling liners that are commonly used in aeroacoustic applications.

Positivity of a weakly singular operator and approximation of wave scattering from the sphere

We investigate properties of a family of integral operators ℬ with a weakly singular compactly supported zonal kernel function on the surface S of the unit 3D sphere. The support is over a spherical cap of height h∈(0,2]. Operators like this arise in some common types of approximations of time domain boundary integral equations (TDBIE) describing the scattering of acoustic waves from the surface of the sphere embedded in an infinite homogeneous medium where h is directly related to the time step size. We show that the Legendre polynomials of degree ℓ≥0 satisfy ∫0h Pℓ(1−z2∕2)dz>0 for all h∈(0,2] and, using spherical harmonics and the Funk–Hecke formula for the eigenvalues of ℬ, that this is a key to unlocking positivity results for a subfamily of these operators. As well as positivity results we give detailed upper and lower bounds on the eigenvalues of ℬ and on ∫S u(x)(ℬu)(x)dx. We give various examples of where these results are useful in numerical approximations of the TDBIE on the sphere and show that positivity of ℬ is a necessary condition for these approximation schemes to be well defined. We also show the connection between the results for eigenvalues and the separation of variables solution of the TDBIE on the sphere. Finally we show how this relates to scattering from an infinite flat surface and Cooke’s 1937 result ∫0r J0(z)dz>0 for all r>0.

Analysis of dynamic crack propagation in two-dimensional elastic bodies by coupling the boundary element method and the bond-based peridynamics

A novel method for predicting dynamic crack propagation based on coupling the boundary element method (BEM) and bond-based peridynamics (BBPD) is developed in this work. The special feature of this method is that it can take full advantages of both the BEM and PD to achieve a higher level of accuracy and efficiency. Based on the scale of the structure and the location of cracks, the considered domain can be divided into a non-cracked region and a cracked region. For the non-cracked region, a meshfree boundary-domain integral equation method (meshfree-BDIEM) is employed in the analysis to reduce the dimension by one and to increase the computational efficiency. The bond-based PD is applied to simulate the cracked region, which can model the initiation and propagation of the cracks automatically. The boundary nodes from the BEM on the interfaces can interact with the material points from the PD directly. By using the displacement continuity and force equilibrium conditions on the interfaces, a combined model is obtained by merging the mass, stiffness, and force matrices from each region of the domain. Both the implicit and explicit BEM-PD coupled solution methods can deliver accurate results without inducing the ghost forces. Several benchmark problems of dynamic crack propagation have been modeled by using the method, which demonstrate that the developed BEM-PD coupled approach can be an efficient numerical tool to model the dynamic crack propagation problems.

Time domain boundary integral equations and convolution quadrature for scattering by composite media

We consider acoustic scattering in heterogeneous media with piecewise constant wave number. The discretization is carried out using a Galerkin boundary element method in space and Runge-Kutta convolution quadrature in time. We prove well-posedness of the scheme and providea prioriestimates for the convergence in space and time.

On the analysis and numerics of united and segregated boundary-domain integral equation systems in 2D

The boundary domain integral equation (BDIE) method provides an alternative formulation to a boundary value problem (BVP) with variable coefficient in terms of integral operators defined on the boundary and the domain. In this paper, we apply two variants of the boundary domain integral equation, the united approach and the segregated approach, to the Dirichlet BVP for the steady diffusion equation with variable coefficient in two dimensions. Details on the derivation of such systems as well as equivalence and well-posedness results are provided. Moreover, we present the discretisation of the two integral equation systems and a comparison of the numerical behaviour of the approximated solutions obtained with the segregated approach and the united approach.

On the Well-Posedness of Two Boundary-Domain Integral Equation Systems Equivalent to the Dirichlet Problem for the Stokes System with Variable Viscosity

On the Well-Posedness of Two Boundary-Domain Integral Equation Systems Equivalent to the Dirichlet Problem for the Stokes System with Variable Viscosity

On a boundary-domain integral equation system for the Robin problem for the diffusion equation in non-homogeneous media

Abstract The Robin problem for the diffusion equation in non-homogeneous media partial differential equation is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one applied by Chkadua, Mikhailov, Natroshvili. in [4]. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed.

Fast Boundary-Domain Integral Method with the [formula omitted]-matrix formulation for large scale numerical investigations

In engineering, several physical models result in inhomogeneous partial differential equations. A prototype of such an equation is the modified Helmholtz equation or also called Yukawa equation. It may result from fluid mechanics (false transient approach) or heat transfer if a semi-discretisation in time with a finite difference schema is applied. Using the Boundary Element Method for the numerical solution of such problems requires to solve a boundary-domain integral equation. The main drawback of all boundary element methods is the quadratic complexity, which exists as well for boundary-domain element methods.Here, a fast approach based on the H2-concept is proposed. The focus is on the discretisation of the domain integral. Respective cluster trees for the domain and the boundary nodes are established. The integral kernels in admissible blocks are approximated with Lagrange interpolation. Further, a recompression is applied, which is here performed with a fully pivoted adaptive cross approximation. The numerical results show that the memory used to store the approximated matrices is logarithmic linear. Considering the matrix formulation of the integral kernel approximation one can reduce the storing space needed in memory to linear complexity.

Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP in 2D with general right-hand side

The mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second-order scalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain is considered. The PDE on the right-hand side belongs to H−1(Ω) or H˜−1(Ω), when neither classical nor canonical conormal derivatives of solutions are well defined. The two-operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of boundary-domain integral equations (BDIEs). Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials, and hence the unique solvability of BDIEs. The equivalence of the BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the corresponding Sobolev spaces.

Two‐operator boundary‐domain integral equation systems for variable‐coefficient mixed Boundary Value Problem in 2D

In this paper, the mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second‐order scalar elliptic differential equation with variable coefficients in a bounded two‐dimensional domain is considered. The two‐operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of two‐operator boundary‐domain integral equations (BDIEs). Although the theory of two‐operator BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix‐based integral layer potentials and hence the unique solvability of BDIEs. The equivalence of the two‐operator BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the appropriate Sobolev spaces.

Analytically-integrated radial integration PBEM for solving three-dimensional steady heat conduction problems

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.

IG-DRBEM of three-dimensional transient heat conduction problems

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.

RI-IGABEM for 2D viscoelastic problems and its application to solid propellant grains

The isogeometric boundary element method (IGABEM) has a broad application prospect due to its exact geometric representation, excellent field approximation and only boundary discretization property. In this paper, IGABEM based on radial integration method (RI-IGABEM) is used for viscoelastic analysis of solid propellant grain. The memory stress, as the initial stress, leads to the boundary-domain integral equations and thus eliminates the only boundary discretization advantage of boundary element method (BEM). The radial integration method (RIM) is applied to transform the domain integral into an equivalent boundary integral by means of the applied points. The usage of RIM makes it possible to only store the strains on the applied points. Meanwhile, Prony-series is used to discretize the general integrals and to store the two most recent time-step strains rather than the time-step strains of the entire process. The combination between RIM and Prony-series will help reduce the storage space and computational time. In addition, by using the fundamental solutions for linear elastic problems and the regularized technologies, the singular integrals can be solved through the previous methods, such as the Telles scheme and element sub-division method. In order to validate the accuracy and robustness of RI-IGABEM in viscoelastic analysis, the influence of the number and position of applied points as well as the time interval on viscoelastic analysis is discussed through comparing with cell discretization methods. A set of numerical examples demonstrates the ability of the scheme to simulate the viscoelastic problems.

Boundary‐domain integral equation systems to the mixed BVP for compressible Stokes equations with variable viscosity in 2D

In this paper, the boundary‐domain integral equations (BDIEs) for the mixed boundary value problem for a compressible Stokes system of partial differential equations with variable coefficient in 2D is considered. An appropriate parametrix is used to reduce this problem to the BDIEs. Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the spaces to ensure the invertibility of corresponding parametrix‐based integral layer potentials and hence the unique solvability of BDIEs. The properties of corresponding potential operators are investigated. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems in appropriate Sobolev spaces are proved.