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Physics-informed neural network (PINN) has gained wide attention for solving forward and inverse problems of partial differential equation (PDE) from both data-driven and model-driven perspectives. In a short time, various machine learning methods based on PINN have been developed for solving a broad range of PDE. To improve the training accuracy of PINN, the popular approach is to introduce penalty parameters to correct the imbalance among different parts of loss function during model training. However, this approach generally fails to eliminate the model error generated by the boundary condition. To eliminate this issue, we propose a gradient auxiliary physics-informed neural network (GA-PINN) for nonlinear biharmonic equation. The key idea of GA-PINN is to split original biharmonic equation subjected to clamped or simply supported boundary conditions into third order or second order differential equations by introducing gradient auxiliary functions. As a consequence, we can rewrite clamped or simply supported boundary conditions as Dirichlet boundary conditions, then conveniently construct the neural network composite functions to satisfy those Dirichlet boundary conditions. We introduce the dynamic weight method to automatically balance the contributions of different loss terms during training. The capabilities of the proposed GA-PINN by solving several nonlinear biharmonic problems are demonstrated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents the degenerate kernels for the polyharmonic and poly-Helmholtz partial differential operators in the polar and spherical coordinates. These degenerate kernels are essential if the analytical/semi-analytical solutions and the mathematical degeneracies involving these operators are conducted by the boundary integral equations. In addition, a two-dimensional creeping flow problem is considered to illustrate the use of the degenerate kernels for problems governed by coupled partial differential equations. The coupled governing equations of the creeping flow problems are converted into a biharmonic equation by using the Hörmander linear partial differential operator theory and the Cartesian partial derivatives on the harmonic and biharmonic degenerate kernels are used to obtain the degenerate kernels of two-dimensional creeping flow problems for BIE analyses. Finally, numerical experiments are conducted to study the convergence of the polyharmonic and poly-Helmholtz degenerate kernels. [ABSTRACT FROM AUTHOR]

This paper is concerned with the development of a new compact 9-point stencil, based on integrated-radial-basis-function (IRBF) approximations, for the discretisation of the first biharmonic equation in two dimensions. Derivatives of not only the first order but also the second order and higher are included in the approximations on the stencil. These nodal derivative values, except for the boundary values of the derivative, are directly derived from nodal variable values along the grid lines rather than from the biharmonic equation, and they are updated through iteration. With these features, the double boundary conditions are imposed in a proper way. The biharmonic equation is enforced at grid points near the boundary without any special treatments. More importantly, they enable the IRBF solution to be highly accurate and not influenced by the RBF width. There is no need for searching the optimal value of the RBF width. The proposed stencil can be used to solve the biharmonic problem defined on a rectangular/non-rectangular domain. A fast convergence rate with respect to grid refinement (up to ten) is achieved. • This paper presents a new compact 9-point stencil for solving the first biharmonic equation. • The approximations on the stencil are constructed using integrated RBFs. • Not only the first derivative but also second derivative and higher ones are included in the approximations. • The RBF solution is highly accurate and not influenced by the RBF width. • A fast convergence rate with respect to grid refinement (up to 10) is achieved. [ABSTRACT FROM AUTHOR]

In this paper, a meshless discrete scheme based on the generalized finite difference method (GFDM) is proposed to solve the biharmonic interface problem. This scheme turns the interface problem to be two non-interface subproblems coupled with the interface conditions. The interface conditions act as the boundary conditions of the subproblems because the interface plays the role of their boundary. Therefore, the proposed method has the advantage of dealing with problems with complex geometrical interfaces. Due to the GFDM approximates the derivatives of the unknown variables by using a linear combination of the values for nearby nodes, the proposed method also has the advantage in dealing with the interface condition with the jump of derivatives. Four numerical examples are provided to verify the accuracy and the stability of the GFDM for biharmonic interface problems. The numerical results show that the H 1 errors can reach almost the same convergence rate of the L ∞ and L 2 errors. The convergence rates are almost and even higher than fourth-order when the fourth-order Taylor Series expansion is adopted in the GFDM. [ABSTRACT FROM AUTHOR]

The second moment of inertia of a continuous axisymmetric object with an arbitrary shape made of a nonhomogeneous material is usually calculated by dividing it into small domains. However, it is a burdensome process to specify the density of the small domains. In this paper, a technique of easily calculating the second moment of inertia of an axisymmetric nonhomogeneous material using boundary integral equations is proposed. The calculations of the mass, primary moment, and center of mass of an arbitrarily shaped object made of a nonhomogeneous material are also shown. A formulization of the boundary element method is utilized, and a technique for the direct numerical integration of the axisymmetric domain using an axisymmetric interpolation method without the need to carry out domain division is proposed. Heterogeneous materials include laminated material composites, in which the density distribution is discontinuous. Axisymmetric interpolation using harmonic and biharmonic functions has a weak point that can be easily overcome using a scale method. [ABSTRACT FROM AUTHOR]

In this study, we develop an approximate formulation for a generalized form of the biharmonic problem based on pseudospectral meshless radial point interpolation (PSMRPI). The boundary conditions are considered as simply supported or clamped, with application to the theory of static analysis of thin-plates. The rigorous steps to analyze such problem are defining the high order derivatives, implementing multiple boundary conditions especially when the geometry of the domain of the problem is complex. In PSMRPI method the nodal points do not need to be regularly distributed and can even be quite arbitrary. It is easy to have high order derivatives of unknowns in terms of the values at nodal points by constructing operational matrices. Furthermore, it is observed that the multiple boundary conditions can be imposed by applying PSMRPI on nodal points near the boundaries of the domain. The main results on the generalized biharmonic problem are demonstrated by some examples to show the validity and trustworthiness of PSMRPI technique. Also, a comparison with the previously standard studied method for the biharmonic problem is done. [ABSTRACT FROM AUTHOR]

In this paper, we propose an improved Generalize Finite Integration Method with Volterra operator to solve multi-dimensional biharmonic equations. The novelty of the proposed method is that its m th order integration matrix will be exactly formulated by using Volterra operator. This leads to a significant improvement in the accuracy of the approximation. Furthermore, the original strict requirement of uniformly distributed collocation points is waived in the proposed method due to the use of piecewise polynomials. This contributes to another distinct advantage in placing collocation points for problems with geometrically complex domains. Four numerical examples are constructed to demonstrate these advantages. Comparisons on the accuracies obtained from quadrature formulas with and without Volterra operator are given. Numerical results indicate that the proposed method is capable for solving multi-dimensional biharmonic equations with superior efficiency, higher accuracy and fewer memory usage. [ABSTRACT FROM AUTHOR]

Solving fourth or higher order differential equations using localized numerical methods as the finite element, the finite difference and the localized radial basis functions (LRBFs), show difficulties to get accurate results. So many authors adopted iterative methods to deal with biharmonic equation by decoupling it into two Poisson's problems. In this paper we investigate the formulation of a mixed fourth order boundary value problem as an optimal control one. Then, we establish a new iterative method by coupling an optimization iterative scheme and localized radial basis functions meshless collocation method to deal with the numerical solution of such problem. To transform the problem into an optimal control one, we firstly construct the constraints functions by splitting the biharmonic equation into two coupled Laplace equations. The Neumann boundary condition is used as energy-like error functional to be minimized. Theoretical analysis of the existence and uniqueness of the solution of such formulated optimization problem and its equivalence to the initial biharmonic problem are also demonstrated. Finally we show the effectiveness of the proposed method by solving problems in both convex and non-convex regular and irregular domain. [ABSTRACT FROM AUTHOR]

Abstract The localized method of fundamental solutions (LMFS) is proposed in this paper for solving two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complicated domains. Traditionally, the method of fundamental solutions (MFS) is a global method and the resultant matrix is dense and ill-conditioned. In this paper, it is the first time that the LMFS, the localized version of the MFS, is proposed. In the LMFS, the solutions at every interior node are expressed as linear combinations of solutions at some nearby nodes, while the numerical procedures of MFS are implemented within every local subdomain. The satisfactions of governing equation at interior nodes and boundary conditions at boundary nodes can yield a sparse system of linear algebraic equations, so the numerical solutions can be efficiently acquired by solving the resultant sparse system. Six numerical examples are given to demonstrate the effectiveness of the proposed LMFS. [ABSTRACT FROM AUTHOR]

The method of fundamental solutions (MFS) has been known as an effective boundary meshless method for solving homogeneous differential equations with smooth boundary conditions and boundary shapes. Despite many attractive features of the MFS, the determination of the source location and the boundaries with sharp corners still pose a certain degree of challenges. In this paper, we revisit another powerful boundary method, the method of approximate fundamental solutions (MAFS), which approximates the fundamental solution using trigonometric functions. In the MAFS, the fundamental solutions for various governed equations can be easily constructed. The placement of the source points is also simple. In this paper, we will apply the MAFS for solving the Laplace equation with non-harmonic boundary conditions and the biharmonic equation with non-biharmonic boundary conditions with highly irregular or non-smooth domains. We will compare the performance of the MAFS and the MFS in these types of problems. [ABSTRACT FROM AUTHOR]

In this paper, the meshless singular boundary method (SBM) is first presented for the numerical solution of biharmonic problems. The SBM uses the source intensity factor to isolate the singularity of fundamental solutions and an inverse interpolation technique (IIT) to compute the source intensity factor, and thus avoids the fictitious boundary required in the method of fundamental solutions. However, the IIT may worsen the performance of the SBM. Then, a further development of the improved SBM (ISBM) is also presented for boundary-only analysis of biharmonic problems. In the ISBM, a new regularization technique is developed to compute the source intensity factor, which leads to higher computational precision and better numerical stability. Some numerical examples illustrate the efficiency and accuracy of both the SBM and the ISBM. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, the inverse Cauchy problems of Laplace equation and biharmonic equation are transformed, by using the method of fundamental solutions (MFS) and the Trefftz method (TM), to the systems of linear equations for determining the expansion coefficients. Then, we propose three different conditioners together with the conjugate gradient method (CGM) to solve the resultant ill-posed linear systems. They are the post-conditioning CGM and the pre-conditioning CGM based on the idea of equilibrated norm for the conditioned matrices, as well as a minimum-distance conditioner. These algorithms are convergent fast and accurate by solving the inverse Cauchy problems under random noise. [Copyright &y& Elsevier]

Abstract: In this paper we numerically solve both the direct and the inverse Cauchy problems of biharmonic equation by using a multiple-scale Trefftz method (TM). The approximate solution is expressed to be a linear combination of T-complete bases, and the unknown coefficients are determined to satisfy the boundary conditions, by solving a resultant linear equations system. We introduce a better multiple-scale in the T-complete bases by using the concept of equilibrated norm of the coefficient matrix, such that the explicit formulas of these multiple scales can be derived. The condition number of the coefficient matrix can be significantly reduced upon using these better scales; hence, the present multiple-scale Trefftz method (MSTM) can effectively solve the inverse Cauchy problem without needing of the overspecified data, which is an incomplete Cauchy problem. Numerical examples reveal the efficiency that the new method can provide a highly accurate numerical solution even the problem domain might have a corner singularity, and the given boundary data are subjected to a large random noise. [Copyright &y& Elsevier]

Abstract: In this paper, analytical particular solutions of multiquadrics and Chebyshev polynomials associated with problems of three-dimensional thermoelasticity are derived, which provides a supplement to the review article of Cheng, Chen, Golberg and Rashed (Engineering Analysis with Boundary Elements 25 (2001), 377). In the derivation, the three coupled second-order partial differential equations (PDEs) are converted into a biharmonic equation. Then, the multiquadric and polynomial particular solutions of the biharmonic equation are obtained respectively by straight integrations and referring to the first author‘s recent study. For the multiquadric particular solutions, they are set to be infinitely differentiable by suitably arranging the coefficients of its Laurent series. And for the polynomial particular solutions, they can be represented explicitly and implemented without any book keeping. Numerical experiments are carried out to validate the derived particular solutions. [Copyright &y& Elsevier]

Abstract: A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed. [Copyright &y& Elsevier]

Abstract: This paper reports a new method, Chebyshev tau meshless method based on the integration–differentiation (CTMMID) for numerically solving Biharmonic-type equations on irregularly shaped domains with complex boundary conditions. In general, the direct application of Chebyshev spectral method based on differentiation process to the fourth order equations leads to the corresponding differentiation matrix with large condition numbers. From another aspect, the strategy based on the integral formula of a Chebyshev polynomial could not only create sparsity, but also improve the accuracy, however it requires a lot of computational cost for directly solving high order two-dimensional problems. In this paper, the construction of the Chebyshev approximations is to start from the mix partial derivative rather than the unknown function . The irregular domain is embedded in a domain of rectangle shape and the curve boundary can be efficiently treated by CTMMID. The numerical results show that compared with the existing results, our method yields spectral accuracy, and the main distinguishing feature is reducing the condition number of fourth order equations on rectangle domain from to . It also appears that CTMMID is effective for the problems on irregular domains. [Copyright &y& Elsevier]

Abstract: A method is proposed for the construction of Green''s functions for the Sophie Germain equation in regions of irregular shape with mixed boundary conditions imposed. The method is based on the boundary integral equation approach where a kernel vector function B satisfies the biharmonic equation inside the region. This leads to a regular boundary integral equation where the compensating loads and moments are applied to the boundary. Green''s function is consequently expressed in terms of the kernel vector function B , the fundamental solution function of the biharmonic equation, and kernel functions of the inverse regular integral operators. To compute moments and forces, the kernel functions are differentiated under the integral sign. The proposed method appears highly effective in computing both displacements and stress components. [Copyright &y& Elsevier]

Abstract: Two numerical methods for the Cauchy problem of the biharmonic equation are proposed. The solution of the problem does not continuously depend on given Cauchy data since the problem is ill-posed. A small noise contained in the Cauchy data sensitively affects on the accuracy of the solution. Our problem is directly discretized by the method of fundamental solutions (MFS) to derive an ill-conditioned matrix equation. As another method, our problem is decomposed into two Cauchy problems of the Laplace and the Poisson equations, which are discretized by the MFS and the method of particular solutions (MPS), respectively. The Tikhonov regularization and the truncated singular value decomposition are applied to the matrix equation to stabilize a numerical solution of the problem for the given Cauchy data with high noises. The L-curve and the generalized cross-validation determine a suitable regularization parameter for obtaining an accurate solution. Based on numerical experiments, it is concluded that the numerical method proposed in this paper is effective for the problem that has an irregular domain and the Cauchy data with high noises. Furthermore, our latter method can successfully solve the problem whose solution has a singular point outside the computational domain. [Copyright &y& Elsevier]

Abstract: The regular hybrid boundary node method (RHBNM) is a new technique for the numerical solutions of the boundary value problems. By coupling the moving least squares (MLS) approximation with a modified functional, the RHBNM retains the meshless attribute and the reduced dimensionality advantage. Besides, since the source points of the fundamental solutions are located outside the domain, ‘boundary layer effect’ is also avoided. However, an initial restriction of the present method is that it is only suitable for the problems which the governing differential equation is in second order. Now, a new variational formulation for the RHBNM is presented further to solve the biharmonic problems, in which the governing differential equation is in fourth order. The modified variational functional is applied to form the discrete equations of the RHBNM. The MLS is employed to approximate the boundary variables, while the domain variables are interpolated by a linear combination of fundamental solutions of both the biharmonic equation and Laplace’s equation. Numerical examples for some biharmonic problems show that the high accuracy with a small node number is achievable. Furthermore, the computation parameters have been studied. They can be chosen in a wide range and have little influence on the results. It is shown that the present method is effective and can be widely applied in practical engineering. [Copyright &y& Elsevier]

Abstract: A Galerkin boundary node method (GBNM) is developed in this paper for solving biharmonic problems. The GBNM combines an equivalent variational form of boundary integral formulations for governing equations with the moving least-squares approximations for construction of the trial and test functions. In this approach, only a nodal data structure on the boundary of a domain is required. In addition, boundary conditions can be implemented accurately and the system matrices are symmetric. The convergence of this method and numerical examples are given to show the efficiency. [Copyright &y& Elsevier]

Abstract: The stick-slip problem is a two-dimensional Stokes flow problem, and is classified into biharmonic equation with crack singularities. The collocation Trefftz method (CTM) is used to provide the very accurate solutions and leading coefficients. In this paper, the error analysis is made, to show the exponential convergence rates, and the new stability analysis is explored more in detail. We derive the bounds of effective and traditional condition numbers, to have the polynomial and the exponential growth rates, respectively. The moderate effective condition number is a suitable criterion of stability for the CTM solution of the stick-slip problem, while the huge condition number is misleading. Besides, numerical experiments are carried out to support the stability analysis. Hence the effective condition number becomes a new trend of stability for numerical partial differential equations. [Copyright &y& Elsevier]

Abstract: Differential quadrature Trefftz method (DQTM) is developed to deal with plate problems defined in irregular domains. DQTM divides the solution into two parts, a particular solution for inhomogeneous biharmonic equation and the general solution for homogeneous biharmonic equation. For the former, differential quadrature method based on the interpolation of the highest derivative (DQIHD) is involved. For the latter, polynomial basis functions are adopted instead of fundamental solutions. We will show that DQTM not only keeps the advantages of traditional differential quadrature method (DQM), high efficiency and accuracy, but also has no difficulties to deal with geometrically irregular domains. [Copyright &y& Elsevier]

Abstract: In this paper, we apply the effective condition number in Huang and Li [Effective condition number and superconvergence of Trefftz method coupled with high order FEM for singularity problems, Eng Anal Bound Elem 2006;30:270–83] to the penalty Trefftz methods (TMs) with high order elements such as Adini''s elements for Poisson''s equations with singularities. The best superconvergence rates in norm can be reached, where h is the maximal boundary length of elements. For the penalty and the penalty plus hybrid Trefftz methods, the bounds of effective condition numbers retain the same. However, for the penalty TM combination, not only are the algorithms simple, but also the numerical solutions are more accurate. Hence we prefer the penalty TM combination to the penalty plus hybrid TM combination. Such a recommendation is based on the effective condition number, but not on the traditional condition number, based on which the penalty TM combination was declined in Li and Yan [Global superconvergence for blending surfaces by boundary penalty plus hybrid FEMs for biharmonic equations. Appl Numer Math 2001;39:61–85]. The new stability analysis in this paper by means of effective condition number may re-evaluate the existing numerical methods; this paper reports the results of Motz''s problem. Note that in Huang and Li [Effective condition number and superconvergence of Trefftz method coupled with high order FEM for singularity problems, Eng Anal Bound Elem 2006;30:270–83], no stability analysis was provided. [Copyright &y& Elsevier]

Abstract: The solution of the equations governing the steady incompressible slow viscous fluid flow is analysed using a novel technique based on a Laplacian decomposition instead of the more traditional approaches based on the biharmonic streamfunction formulation or the velocity-pressure formulation. This results in the need to solve the Laplace equations for the pressure and other auxiliary harmonic functions which arise from the ideas of Almansi''s decomposition. These equations, which become coupled through the boundary conditions, are numerically solved using the boundary element method (BEM). Results both on the boundary and inside the solution domain are presented and discussed for a simple benchmark test example and a few applications in smooth and non-smooth geometries in order to illustrate that the Laplacian decomposition in combination with BEM provides an efficient technique, in terms of accuracy and convergence, to investigate numerically a Stokes flow. [Copyright &y& Elsevier]

Abstract: This paper reports a new boundary-integral-equation method (BIEM) for numerically solving biharmonic problems with Dirichlet boundary conditions. For the solution of these problems in convex polygons, it was found that the accuracy of the conventional BIEM is significantly reduced, and spurious oscillatory behaviour is often observed in the boundary solutions especially for areas near corners (Mai-Duy N, Tanner RI. An effective high order interpolation scheme in BIEM for biharmonic boundary value problems. Eng Anal Bound Elem 2005;29:210–23). In this study, a new treatment for these difficulties is proposed. The unknown functions in boundary integrals are approximated using a domain-type interpolation scheme rather than traditional boundary-type interpolation schemes. Two test problems are considered to validate the formulation and to demonstrate the attractiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: This paper reports a new indirect radial-basis-function (RBF) collocation method for numerically solving biharmonic boundary-value problems. The RBF approximations are constructed through an integration process. The new feature here is that integration constants are eliminated pointwise using the prescribed boundary conditions rather than employed as network weights. This treatment of integration constants is particularly well suited for the case of discretizing the governing differential equations on a set of scattered points. The proposed method is verified through the solution of benchmark test problems. Accurate results and high convergence rates are obtained. [Copyright &y& Elsevier]

Abstract: This paper presents an effective high order boundary integral equation method (BIEM) for the solution of biharmonic equations. All boundary values including geometries are approximated by high order radial basis function networks (RBFNs) rather than the conventional low order Lagrange interpolation schemes. For a better quality of approximation, the networks representing the boundary values and their derivatives are constructed by using integration processes. Prior conversions of network weights into nodal variable values are employed in order to form a square system of equations. Numerical results show that the proposed BIEM attains a great improvement in solution accuracy, convergence rate and computational efficiency over the linear- and quadratic-BIEMs. [Copyright &y& Elsevier]