Boundary Knot Research Articles

Boundary Knots Method with ghost points for high-order Helmholtz-type PDEs in multiply connected domains

This paper proposes the Boundary Knot Method with ghost points (BKM-G), which enhances the performance of the BKM for solving 2D (3D) high-order Helmholtz-type partial differential equations in domains with multiple cavities. The BKM-G differs from the conventional BKM by relocating the source points from the boundary collocation nodes to a random region, such as a circle (sphere) encompassing the original domain in 2D (3D). Compared with classical BKM, this modification improves accuracy without sacrificing simplicity and efficiency. Moreover, this paper investigates and analyzes the effect of the ghost circle/sphere’s radius R in BKM-G for solving various high-order Helmholtz-type PDEs. Numerous 2D and 3D numerical examples illustrate that the BKM-G outperforms the BKM for a wide range of R. The effectiveness of the proposed effective condition number (ECN) approach in finding the optimal R has also been demonstrated. Furthermore, the economic ECN (EECN) is studied to significantly improve the efficiency of ECN.

Simplified boundary knots method with ghost points for high-order harmonic-type PDEs

Simplified boundary knots method with ghost points for high-order harmonic-type PDEs

An improved localized boundary knot method for 3D acoustic problems

In this paper, an improved localized boundary knot method (LBKM) is proposed for 3D acoustic problems. The domain is divided into many small subdomains, and the solutions at the nodes of each subdomain are approximated by using the linear combination of the non-singular general solutions in the LBKM. However, the interpolant matrix generation for each subdomain in the original LBKM is associated with the variables at both domain and boundary nodes. In the present method, only the variables at domain nodes are considered for the matrix generation, and the boundary conditions are directly imposed during the matrix generation. As a consequence, the computational efficiency of the LBKM should be raised. Two examples are provided to illustrate the performance of the present method.

The local boundary knots method for solution of Stokes and the biharmonic equation

The paper focuses on the derivation of a local variant of the boundary knot method (BKM) for the cases of Stokes flow and the biharmonic equation. Compared to the global variant, the local one leads to a sparse result matrix of the system of equations and thus makes the solution of especially large-scale problems more efficient. It is also important to keep the conditionality of the interpolation matrix within reasonable bounds. For the localization, a combination of BKM and finite collocation method was used. The results of the local variant were compared on several examples and the dependence of the solution on the density of the point network and the dimensions of the stencil used was also tested in the paper.

Modified Boundary Knot Method for Multi-Dimensional Harmonic-Type Equations

This paper presents the modified boundary knot method (MBKM) for solving the homogeneous harmonic type boundary value problems (BVPs). Since no non-singular general solutions are applicable for harmonic-type equations, the general solutions of Helmholtz-type operator with a free parameter [Formula: see text] can be used to approximate the solutions of these problems by adjusting the [Formula: see text]. Compared with the classical boundary knot method (BKM) where the source nodes are the same as the boundary collocation nodes, the MBKM employs the ghost points method, which resets the source points to a disk-like region covering the primary problem area. This modification results in better accuracy without any increase in the computational cost. On the other hand, as the accuracy of the MBKM depends heavily on the parameter [Formula: see text], the effective condition number (ECN) is first employed to find a proper [Formula: see text] in MBKM. Several 2D and 3D numerical examples are listed to illustrate the superior performance of the MBKM in solving harmonic-type BVPs. The accuracy of MBKM is improved by one to two orders of magnitude compared to the classical BKM. Meanwhile, the validity of the ECN for obtaining a suitable [Formula: see text] for problems under complex geometric regions is also demonstrated.

The enhanced boundary knot method with fictitious sources for solving Helmholtz-type equations

The boundary knot method (BKM) is a boundary meshless method for solving homogeneous boundary value problems. The fact that the BKM uses non-singular general solution without the need of source points outside the domain is at the expense of the accuracy when compared to the well-known method of fundamental solutions. In this paper, a new approach called the enhanced BKM (EBKM) is proposed to greatly improve the accuracy of the BKM by introducing fictitious sources which can be placed inside and/or outside the domain quite freely. We also adopt various techniques to place these fictitious sources in an optimal way.

Linear-Time Algorithm for Computing the Bernstein–Bézier Coefficients of B-spline Basis Functions

A new differential-recurrence relation for the B-spline functions of the same degree is proved. From this relation, a recursive method of computing the coefficients of B-spline functions of degree m in the Bernstein-Bézier form is derived. Its complexity is proportional to the number of coefficients in the case of coincident boundary knots. This means that, asymptotically, the algorithm is optimal. In other cases, the complexity is increased by at most O ( m 3 ) . When the Bernstein-Bézier coefficients of B-spline basis functions are known, it is possible to compute any B-spline function in linear time with respect to its degree by performing the geometric algorithm proposed recently by the authors. This algorithm scales well when evaluating the B-spline curve at multiple points, e.g., in order to render it, since one only needs to find the coefficients for each knot span once. When evaluating many B-spline curves at multiple points (as is the case when rendering tensor product B-spline surfaces), such approach has lower computational complexity than using the de Boor-Cox algorithm. The numerical tests show that the new method is efficient. The problem of finding the coefficients of the B-spline functions in the power basis can be solved similarly. • A new differential-recurrence relation for the B-spline functions of the same degree is proved. • From this relation, a recursive method of computing the coefficients of B-spline functions of degree m in the Bernstein-Bézier form is derived. • Its complexity is proportional to the number of coefficients in the case of coincident boundary knots. This means that, asymptotically, the algorithm is optimal. • When the Bernstein-Bézier coefficients of B-spline basis functions are known, it is possible to compute any B-spline function in linear time with respect to its degree by performing the geometric algorithm proposed recently by the authors. • This algorithm scales well when evaluating many B-spline curves with the same knot sequences at multiple points, e.g., when rendering tensor product B-spline surfaces. • When evaluating many B-spline curves at multiple points, such approach has lower computational complexity than using the de Boor-Cox algorithm. • The numerical tests show that the new method is efficient. • The problem of finding the coefficients of the B-spline functions in the power basis can be solved similarly.

The Modified Local Boundary Knots Method for Solution of the Two-Dimensional Advection–Diffusion Equation

This paper deals with a new modification of the local boundary knots method (LBKM), which will allow the irregular node distribution and the arbitrary shape of the solution domain. Unlike previous localizations, it has no requirements on the number of nodes in the support or on the number of virtual points. Owing to the limited number of virtual points, the condition number of boundary knots matrix remains relatively low. The article contains the derivation of the relations of the method for steady and unsteady states and shows its effectiveness in three control examples.

A boundary meshless method for dynamic coupled thermoelasticity problems

This work develops the boundary knot method (BKM) for two-dimensional (2D) coupled thermoelasticity problems in the frequency domain. By taking the non-singular general solution satisfying the governing equations as the basis function, the BKM does not require domain discretization. Nevertheless, the non-singular solution for the considered problems is absent. The Helmholtz decomposition and eigen-analysis are introduced to firstly derive the non-singular general solution, and thus the BKM can be formulated in terms of the discretized boundary points. In the final, the exponential window method (EWM) in conjunction with the BKM is employed to perform the transient analysis. The accuracy and feasibility of the proposed method are presented via numerical examples.

Local knot method for solving inverse Cauchy problems of Helmholtz equations on complicated two‐ and three‐dimensional domains

AbstractThis article presents a local knot method (LKM) to solve inverse Cauchy problems of Helmholtz equations in arbitrary 2D and 3D domains. The Moore–Penrose pseudoinverse using the truncated singular value decomposition is employed in the local approximation of supporting domain instead of the moving least squares method. The developed approach is a semi‐analytical and local radial basis function collocation method using the non‐singular general solution as the basis function. Like the traditional boundary knot method, the LKM is simple, accurate and easy‐to‐program in solving inverse Cauchy problems associated with Helmholtz equations. Unlike the boundary knot method, the new scheme can directly reconstruct the unknowns both inside the physical domain and along its boundary by solving a sparse linear system, and can achieve a satisfactory solution. Numerical experiments, involving the complicated geometry and the high noise level, confirm the accuracy and reliability of the proposed method for solving inverse Cauchy problems of Helmholtz equations.

Localized Boundary Knot Method for Solving Two-Dimensional Inverse Cauchy Problems

In this paper, a localized boundary knot method is adopted to solve two-dimensional inverse Cauchy problems, which are controlled by a second-order linear differential equation. The localized boundary knot method is a numerical method based on the local concept of the localization method of the fundamental solution. The approach is formed by combining the classical boundary knot method with the localization method. It has the potential to solve many complex engineering problems. Generally, in an inverse Cauchy problem, there are no boundary conditions in specific boundaries. Additionally, in order to be close to the actual engineering situation, a certain level of noise is added to the known boundary conditions to simulate the measurement error. The localized boundary knot method can be used to solve two-dimensional Cauchy problems more stably and is truly free from mesh and numerical quadrature. In this paper, the stability of the method is verified by using multi-connected domain and simply connected domain examples in Laplace equations.

Zero Set Structure of Real Analytic Beltrami Fields

In this paper, we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected 3-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected 1-dimensional submanifolds with (possibly empty) boundary and tame knots. Further, we consider the question of how complicated these tame knots can possibly be. To this end, we prove that on the standard (open) solid toroidal annulus in {mathbb {R}}^3, there exist for any pair (p, q) of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric, there exists a real analytic Beltrami field, corresponding to the eigenvalue +1 of the curl operator, whose zero set is precisely given by a standard (p, q)-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.

Analysis of 2D heat conduction in nonlinear functionally graded materials using a local semi-analytical meshless method

<abstract> <p>This paper proposes a local semi-analytical meshless method for simulating heat conduction in nonlinear functionally graded materials. The governing equation of heat conduction problem in nonlinear functionally graded material is first transformed to an anisotropic modified Helmholtz equation by using the Kirchhoff transformation. Then, the local knot method (LKM) is employed to approximate the solution of the transformed equation. After that, the solution of the original nonlinear equation can be obtained by the inverse Kirchhoff transformation. The LKM is a recently proposed meshless approach. As a local semi-analytical meshless approach, it uses the non-singular general solution as the basis function and has the merits of simplicity, high accuracy, and easy-to-program. Compared with the traditional boundary knot method, the present scheme avoids an ill-conditioned system of equations, and is more suitable for large-scale simulations associated with complicated structures. Three benchmark numerical examples are provided to confirm the accuracy and validity of the proposed approach.</p> </abstract>

Localized boundary knot method for 3D inhomogeneous acoustic problems with complicated geometry

This paper reports the first attempt to extend a novel localized boundary knot method to the 2D and 3D acoustic problems with complicated geometry. The solution of original partial differential equation with inhomogeneous term is firstly described as a sum of a particular solution and a homogeneous one. Secondly, the Chebyshev interpolation technique is applied to the approximation of particular solution. The interpolation approach adopts the Chebyshev-Gauss-Lobatto nodes to offer the spectral convergence and high accuracy. And then the numeircal solution of homogeneous equation is obtained by applying the localized boundary knot method with non-singular general solutions. As a localized meshless method, the localized boundary knot method in conjunction with the interpolation technique is very simple mathematically, accurate numerically, and particularly feasible for large-scale computation in a complicated geometry. Numerical experiments including 2D and 3D models verify the performance of the proposed method for solving inhomogeneous Helmholtz-type equations.

Localized Boundary Knot Method for Solving Two-Dimensional Laplace and Bi-Harmonic Equations

In this paper, a localized boundary knot method is proposed, based on the local concept in the localized method of fundamental solutions. The localized boundary knot method is formed by combining the classical boundary knot method and the localization approach. The localized boundary knot method is truly free from mesh and numerical quadrature, so it has great potential for solving complicated engineering applications, such as multiply connected problems. In the proposed localized boundary knot method, both of the boundary nodes and interior nodes are required, and the algebraic equations at each node represent the satisfaction of the boundary condition or governing equation, which can be derived by using the boundary knot method at every subdomain. A sparse system of linear algebraic equations can be yielded using the proposed localized boundary knot method, which can greatly reduce the computer time and memory required in computer calculations. In this paper, several cases of simply connected domains and multi-connected domains of the Laplace equation and bi-harmonic equation are demonstrated to evidently verify the accuracy, convergence and stability of this proposed meshless method.

The boundary knot method for the solution of two-dimensional advection reaction-diffusion and Brusselator equations

PurposeThis study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a semi-implicit scheme in time for finding a new numerical solution of the advection–reaction–diffusion and reaction–diffusion systems in two-dimensional spaces, which arise in biology.Design/methodology/approachFirst, the BKM is applied to approximate the spatial variables of the studied mathematical models. Then, this study derives fully discrete scheme of the studied models using a semi-implicit scheme based on Crank–Nicolson idea, which gives a linear system of algebraic equations with a non-square matrix per time step that is solved by the singular value decomposition. The proposed approach approximates the solution of a given partial differential equation using particular and homogeneous solutions and without considering the fundamental solutions of the proposed equations.FindingsThis study reports some numerical simulations for showing the ability of the presented technique in solving the studied mathematical models arising in biology. The obtained results by the developed numerical scheme are in good agreement with the results reported in the literature. Besides, a simulation of the proposed model is done on buttery shape domain in two-dimensional space.Originality/valueThis study develops the BKM combined with MAEM for solving the coupled systems of (advection) reaction–diffusion equations in two-dimensional spaces. Besides, it does not need the fundamental solution of the mathematical models studied here, which omits any difficulties.

Super-exponential growth rates of condition number in the boundary knot method for the Helmholtz equation

The boundary knot method (BKM) is applied to the Helmholtz equation in 2D bounded simply-connected domains, to show high accuracy of the solutions obtained by not many fundamental solutions (FS) used. When the Bessel function is chosen as the FS, the optimal polynomial convergence rates are obtained for disk domains. Moreover, the bounds of condition number (Cond) are derived for disk domains, to show a super-exponential growth via the number of FS used. The super-exponential growth of Cond is new and intriguing in the numerical methods for partial differential equations (PDE). It is also imperative for the BKM that good numerical solutions may be achieved by balancing accuracy and instability. Numerical experiments are carried out to support the analysis made. Comparisons between the BKM and the MFS are also made.

Local knot method for 2D and 3D convection–diffusion–reaction equations in arbitrary domains

In this paper, a novel local knot method (LKM) is presented to solve the 2D and 3D convection–diffusion–reaction equations in arbitrary domains. Contrary to the traditional boundary knot method, the proposed scheme requires the nodes not only on the boundary but also inside the domain. For each node, we can find a local subdomain containing a certain number of neighboring nodes. Utilizing the non-singular general solution of differential operator and the known boundary conditions, a sparse linear system is established to approximate the solutions at all nodes over the physical domain. The present LKM is a local meshless method with the merits of being mathematically simple, numerically accurate and easy to large-scale computation. Two numerical examples, involving 2D and 3D complicated domains, are provided to illustrate the effectiveness and accuracy of the new methodology.

A boundary knot method for 3D time harmonic elastic wave problems

This paper makes the first attempt to apply the boundary knot method (BKM) to 3D time harmonic elastic wave problems. Unlike the famous method of fundamental solutions (MFS), the BKM employs non-singular general solutions as the basis functions to approximate the solutions. Therefore, the BKM overcomes the auxiliary boundary issue of the MFS, and inherits the advantages of the MFS, i.e., mathematically simple, easy-to-program, meshless, integration-free and accurate. Particularly, the non-singular general solutions of 3D time harmonic elastic wave problems for the BKM are firstly derived in this study. Two examples are carried out to verify the accuracy and convergence of the BKM.

Localized boundary knot method and its application to large-scale acoustic problems

We present a new localized boundary knot method (LBKM) for the solution of large-scale partial differential equations based on the non-singular general solutions. The proposed local algorithm only requires several discrete nodes inside the physical domain and along its boundary without any mesh. For every node, a local subdomain with a simple geometry can be firstly determined via Euclidean distance between nodes. And then the unknown variable at every node can be expressed as a linear combination of function values at nodes inside its corresponding local subdomain. Finally, a sparse linear system can be formed by using the governing equation and the corresponding boundary conditions. In our computations, two different-types of LBKM formulations have been proposed for deriving the final sparse linear system. One is based on the inverse matrix calculation and the other is based on the moving least square (MLS) technique. Unlike the traditional boundary knot method (BKM) with the “global” discretization, the present LBKM is a local discretization method and requires less computer time and storage due to its feature of sparse and banded matrix, which makes it more suitable for solving large-scale problems. Preliminary numerical experiments are carried out in the large-scale acoustics models and bimaterial problems. The comparisons of proposed two types of LBKM formulations are also investigated. The results show the accuracy, validity and great promising applications of the proposed LBKM for the simulation of large-scale problems in complicated geometries.