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In this paper, an approach to improve the method of angular basis function (MABF) proposed by Young et al. (2015) is proposed. Instead of using lnr in the method of fundamental solution (MFS), the MABF adopts θ to construct the solution. However, since the nature of θ introduces multiple values along the branch cut such that to avoid the branch cut passing through the domain is an important issue (Li et al., 2018). Noticing this difficulty, Alves et al. (2018) first proposed a remedy which used a pair of two points to restrict the discontinuity appearing only along the line segment between two points, and they named this approach as cracklets. In this article, the two-point angular basis function (cracklets) is investigated deeply. We explain why for a multiply connected domain with a logarithm singular solution the cracklets will encounter failure. To resolve this difficulty, one can adopt the proposed method (cracklets) with the MFS or one can use domain decomposition method to separate the domain into several singly connected domains. Seven numerical examples are provided to show the validity of this method, and examples for dealing with the multiply connected domain are focused to support our claims.

In this paper, after deriving certain adjoint eigenfunctions as test functions, as well as the bases of unknown pollutant source, the numerical algorithm based on the weak-form integral equation method is developed, of which the expansion coefficients can be found exactly in closed-form, without resorting to numerical solution. The closed-form expansion coefficients method is effective for the inverse pollutant source problem (IPSP), which can recover the unknown pollutant source by means of an extra final time condition. Numerical examples of the IPSP demonstrate that the closed-form expansion coefficients method is stable against large noises up to 10%–50%.

In this paper we first express the wave equation in terms of the Minkowskian polar coordinates and generate a set of complete hyperbolic type Trefftz bases: r k cosh ź ( k ź ) and r k sinh ź ( k ź ) , which are further transformed to wave polynomials as the trial solution bases for the one-dimensional wave equation. In order to stably solve the wave propagation problems long-term we develop a multiple-scale Trefftz method (MSTM), of which the scales are determined a priori by the collocation points. Then we derive a very simple method of multi-dimensional wave polynomials, equipped with different spatial directions which being the normalized wavenumber vectors, as the polynomial Trefftz bases for solving the multi-dimensional wave equations, which is named a multiple-direction Trefftz method (MDTM). Several numerical examples of two- and three-dimensional wave equations demonstrate that the present method is efficient and stable.

The polynomial Trefftz method consists of the polynomial type solutions as bases, providing a cheap boundary-type meshless method to solve the heat conduction equation, since the bases automatically satisfy the governing equation. In order to stably solve the backward heat conduction problem (BHCP), and the inverse heat source problem (IHSP) together with the boundary condition recovery problem by a polynomial Trefftz method, which are both known to be highly ill-posed, we introduce a multiple-scale post-conditioner in the resultant linear system to reduce the condition number. Then the conjugate gradient method (CGM) is used to solve the post-conditioned linear system to determine the unknown expansion coefficients. In the multiple-scale polynomial Trefftz method (MSPTM) the scales are determined a priori by the collocation points on space–time boundary, which can retrieve the missing initial data, the unknown time-dependent heat source as well as the boundary condition rather well. Several numerical examples of the inverse heat conduction problems demonstrate that the MSPTM is effective and accurate, even for those of severely ill-posed inverse problems under very large noises.

In this paper, we recover unknown space–time dependent forces imposed on the vibrating Euler–Bernoulli beams under different boundary conditions, including simply supported, clamped-hinged, cantilevered, and two-end fixed conditions. The data overspecified to recover the external force are the final time displacement and the right-side strain or right-side moment of the beam. We develop a family of boundary shape functions, which automatically satisfy the initial conditions, final time condition, and homogeneous boundary conditions for each type of beam. When the solution is obtained using the method of superimposing boundary shape functions and solving a small-scale linear system to satisfy an extra right-side boundary condition, the unknown force can be recovered through back-substitution of the solution into the Euler–Bernoulli beam equation. The accuracy and robustness of the proposed methods are confirmed by comparing the recovered results of seven examples to the exact forces, even though considerable noise is present in the overspecified data.

The polynomial expansion method is a useful tool to solve partial differential equations (PDEs). However, the researchers seldom use it as a major medium to solve PDEs due to its highly ill-conditioned behavior. We propose a single-scale and a multiple-scale Pascal triangle formulations to solve the linear elliptic PDEs in a simply connected domain equipped with complex boundary shape. For the former method a constant parameter R 0 is required, while in the latter one all introduced scales are automatically determined by the collocation points. Then we use the multiple-scale method to solve the inverse Cauchy problems, which is very accurate and very stable against large noise to 20%. Numerical results confirm the validity of the present multiple-scale Pascal polynomial expansion method.

In this paper, the modified polynomial expansion method is developed to solve problems of identifying the time dependent heat source, in which an inverse problem is encountered. Aimed at this problem, the variation of variables is adopted to eliminate the unknown heat source and obtain a six-line boundary value problem. As compared with the conventional four-line boundary value problem, the six-line boundary value problem is quite hard to be dealt with. After the unknown non-homogeneous term being eliminated, the polynomial expansion method is introduced to discretize the time and space fields, respectively. Then, the distribution of temperature is expressed as a linear superposition of polynomial functions. After that, a characteristic length concept is adopted to resolve the ill-posed matrix problems arising in those conventional polynomial expansion methods. The desired heat source function can be obtained by putting the solution of the six-line boundary value problem into differential operations. Several numerical experiments with designed examples are included to validate the accuracy and effectiveness of the proposed approach.

In this article, the solutions for the Helmholtz equation for forward problems with high wave number and ill-posed inverse problems using the multiple scales Trefftz collocation method are investigated. The resulting linear algebraic systems for these problems are ill-posed and therefore require special treatments. The equilibrated matrix concept is adopted to determine the scales and to construct an equivalent linear algebraic problem with a leading matrix less ill-posed such that standard solver like the conjugate gradient method (CGM) can be adopted. Five examples, including two forward problems with the high wave number and three inverse Cauchy problems, are given to show the validity for the approach. Results show that the equilibrated matrix concept can yield a less ill-posed leading matrix such that the conventional linear algebraic solver like CGM can be successfully adopted. This approach has a very good noise resistance.

This paper proposes the numerical solution of three-dimensional Laplacian problems based on the multiple scale Trefftz method with the incorporation of the dynamical Jacobian-inverse free method. A numerical solution for three-dimensional Laplacian problems was approximated by superpositioning T-complete functions formulated from 18 independent functions satisfying the governing equation in the cylindrical coordinate system. To mitigate a severely ill-conditioned system of linear equations, this study adopted the newly developed multiple scale Trefftz method and the dynamical Jacobian-inverse free method. Numerical solutions were conducted for problems involving three-dimensional groundwater flow problems enclosed by a cuboid-type domain, a peanut-type domain, a sphere domain, and a cylindrical domain. The results revealed that the proposed method can obtain accurate numerical solutions for three-dimensional Laplacian problems, yielding a superior convergence in numerical stability to that of the conventional Trefftz method.

In this paper, the boundary detection problem, which is governed by the Laplace equation, is analyzed by the modified collocation Trefftz method (MCTM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the boundary detection problem, the Cauchy data is given on part of the boundary and the Dirichlet boundary condition on the other part of the boundary, whose spatial position is unknown a priori. By adopting the MCTM, which is meshless and integral-free, the numerical solution is expressed by a linear combination of the T-complete functions of the Laplace equation. The use of a characteristic length in MCTM can stabilize the numerical procedure and ensure highly accurate solutions. Since the coefficients of MCTM and the position of part of the boundary are unknown, to collocate the boundary conditions will yield a system of nonlinear algebraic equations; the ECSHA, which is exponentially convergent, is adopted to solve the system of nonlinear algebraic equations. Several numerical examples are provided to demonstrate the ability and accuracy of the proposed meshless scheme. In addition, the consistency of the proposed scheme is validated by adding noise into the boundary conditions.

This study presents the numerical predictions of the fluid flow characteristics within the recirculation zone for a backward-facing step with uniform normal mass bleed. The turbulent governing equations are solved by a control-volume-based finite-difference method with power-law scheme. A new turbulence model is proposed to describe the turbulent structure. Non-uniform staggered grids are used. The parameters studied include entrance Reynolds number (Re), and the velocity of the normal mass bleed (Vs). The channel expansion ratio ER = 1.3, and the working medium is air. The numerical results show the uniform normal mass bleed suppresses the reverse horizontal velocity, turbulence intensity, and Reynolds shear stress within the recirculation zone. The attachment point extends to downstream. Better computational predictions are obtained with the new turbulence model by the introduction of the Kolmogorov velocity scale, Ue = (ve) 1 4 instead of friction velocity Uv.