Collocation Trefftz Method Research Articles

Localized space-time Trefftz method for diffusion equations in complex domains

This paper introduces an advanced localized space-time Trefftz method tackling boundary value predicaments within complex two-dimensional domains governed by diffusion equations. In contrast to the widespread space-time collocation Trefftz method, which typically produces dense and ill-conditioned matrices, the proposed strategy employs a localized collocation scheme to remove these constraints. In particular, this is beneficial in multi-connected configurations or when dealing with significant variations in field values. To the best of our knowledge, this is the first space-time collocation Trefftz method adaptation, which is referred to as the localized space-time Trefftz method in this paper. The latter combines the space-time collocation Trefftz method principles, which allows to eliminate the need for mesh and numerical quadrature in its application. The localized space-time Trefftz method represents each interior node expressed as a linear blend of its immediate neighbors, while the space-time collocation Trefftz method applies numerical techniques within distinct subdomains. A sparse system of linear algebraic equations with internal points satisfying the governing equation, and boundary points satisfying the boundary conditions, allows to obtain numerical solutions using matrix systems. The localized space-time Trefftz method retains the easy-to-use properties and mesh-free structure of the space-time collocation Trefftz method, and it mitigates its ill-conditioning characteristics. Due to the localization principle and the consideration of overlapping subdomains, the solutions in the proposed localized space-time Trefftz method are more simply and compactly expressed compared with those in the space-time collocation Trefftz method, especially when dealing with multiply-connected domains. Numerical examples for simply-connected and multiply-connected domains are then provided to demonstrate the high precision and simplicity of the proposed localized space-time Trefftz method. The obtained results show that the latter has very high accuracy in solving two-dimensional diffusion problems. Compared with the traditional space-time collocation Trefftz method, the proposed mesh-free strategy yields solutions with higher precision while significantly reducing the instability.

A novel floodwave response model for time-varying streambed conductivity using space-time collocation Trefftz method

Streambed conductivity can vary substantially over short time intervals during flood events, altering groundwater – surface water interactions. Traditional floodwave response models (FRM) developed by linking a unit step response function with the convolution integral are limited by the conditions of a stationary aquifer-stream system and a hydrostatic equilibrium initial condition. However, these two conditions can hardly be satisfied when streambed conductivity changes with time. As a result, in this study a new nonstationary FRM (NFRM) that can consider time-varying streambed conductivity was developed using a space–time collocation Trefftz method. In this new model, the aquifer was assumed homogeneous, and the groundwater flow was simplified to be one-dimensional and perpendicular to a partially penetrating river. The NFRM can be applied to both the forward calculation of groundwater responses to fluctuations of river stage and the inverse calculation of time-varying streambed conductivity. A wide range of synthetic cases generated by the numerical simulations of aquifer-stream interactions was used for validation. Good agreement between the results computed from the NFRM and the synthetic data can be found in all cases (Nash-Sutcliffe model efficiency coefficient > 0.90; Mean Absolute Relative Error < 0.03), showing the accuracy and robustness of the NFRM. The NFRM was then successfully applied to a study site on the Tallahatchie River (Mississippi, USA). The estimated streambed conductivity was found to vary between 10-3 m/d to 10-2 m/d over the studied period. It generally increased during high-stage events and decreased when the river stayed in a low-stage condition, but also generally depended on seepage direction with increases occurring while groundwater discharged to the river and decreases while river water recharged the aquifer.

A numerical analysis of the generalised collocation Trefftz method for some 2D Laplace problems

This paper analyses the generalised collocation Trefftz method which allows to combine the advantages of the T-Trefftz and MFS. The initial idea of the method is to approximate the solution with a linear combination of many basis functions with many source points. The application of only one source point with nonsingular basis function allows for set up linearly independent set. On the other hand, using logarithmic and negative power bases for the source points enables better control over quality of the solution. The validity of the proposed method is conducted for the potential problem in a two-dimensional simply and doubly connected domain without using the domain decomposition.

A Regularized Trefftz Method for Solving an Inverse Problem for the Laplace Equation

The present paper proposes a new regularised Trefftz method to recover the boundary value on a non-accessible boundary of an annulus from overdetermined data on the accessible boundary of that annulus. Considering that the available data have a Fourier expansion, we consider the Tikhonov damping factor in constructing our regularized scheme, which satisfies the boundary value problem. At the same time, the convergence estimates for the regularised solution will be established under an assumption for the exact solution. The finite term truncation of the series expansion allows us to match the boundary condition as accurately as desired. By the collocation method, we derive a system of linear equations that can be uniquely solved to obtain the coefficients and preset the regularisation parameter and the damping factor to ensure adequate stability. The numerical efficiency of the proposed method is investigated with a high truncation number in comparison with the modified collocation Trefftz method.

Численные эксперименты двойственным методом нулевого поля в задаче Дирихле для уравнения Лапласа в эллиптических областях с эллиптическими отверстиями

Dual techniques have been used in many engineering papers to deal with singularity and ill-conditioning of the boundary element method (BEM). In the first part of the two-part article, our efforts were focused on studying the theoretical aspects of this problem, including the analysis of errors and the study of stability. We provided the theoretical analysis for Laplace equation in elliptic domains with elliptic holes. To bypass the degenerate scales of Dirichlet problems, the second and the first kinds of the null field methods (NFM) are used for the exterior and the interior boundaries, simultaneously. This approach is called the dual null field method (DNFM). This paper is the second part of the study. Numerical results for degenerate models of an elliptic domain with one elliptic hole at 𝑎 + 𝑏 = 2 are carried out to verify the theoretical analysis obtained. The collocation Trefftz method (CTM) is also designed for comparisons. Both the DNFM and the CTM can provide excellent numerical performances. The convergence rates are the same but the stability of CTM is excellent and can achieve the constant condition numbers, Cond = 𝑂(1).

An efficient localized Trefftz-based collocation scheme for heat conduction analysis in two kinds of heterogeneous materials under temperature loading

This paper presents a novel localized collocation Trefftz method (LCTM) for heat conduction analysis in two kinds of heterogeneous materials (functionally graded materials and multi-medium materials) under temperature loading. In contrast to the conventional collocation Trefftz method (CTM), the proposed LCTM divides the whole domain into many stencil support domains consisting of several discretization nodes. An efficient technique, the generalized reciprocity method (GRM), is proposed to derive the problem-dependent T-complete functions for approximating the particular solution of the nonhomogeneous equations in the stencil support domains. Based on the moving least square (MLS) technique and T-complete functions, the LCTM numerical differentiation formulation at a certain node can be derived by using a linear combination of the T-complete functions at its adjacent discretization nodes in the related stencil support domain. It inherits the semi-analytical property from the conventional CTM and avoids the ill-conditioned dense matrix problem. Besides, the quadrant criterion is applied to guarantee the stability of the proposed LCTM, and the domain decomposition method (DDM) is introduced to divide the multi-medium domains into several single-medium sub-domains. Numerical results demonstrate the accuracy and efficiency of the proposed LCTM in comparison with the known analytical solutions and the finite element method (FEM) results.

Applications of the Trefftz method to the anti-plane fracture of 1D hexagonal piezoelectric quasicrystals

In terms of solving fracture mechanics problems of quasicrystals, the boundary element method known as Trefftz Method is used for the first time. In this paper, the collocation Trefftz method as an indirect Trefftz method has been demonstrated to be an excellent numerical method for the anti-plane fracture problems in quasicrystals plate, which can exactly satisfy the governing equations and approximately satisfy the boundary conditions. The algorithms of the collocation Trefftz method and the trial functions for the anti-plane fracture problems of piezoelectric quasicrystals are presented. The effects of the geometry size, the loadings and the materials properties on the fracture results are studied. For the materials with three dimensional constitutive relations, such as piezoelectric quasicrystals, the Trefftz method is an effective numerical method to solve the fracture problems.

Singularity problems from source functions as source nodes located near boundaries; numerical methods and removal techniques

Consider the Dirichlet problem for Laplace’s/Poisson’s equation in a bounded simply-connected domain S. The source function qln|PQ*¯| is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by qln|PQ*¯| as the source node Q* is located near the boundary Γ(=∂S). So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node Q* located near Γ. In this paper, an additional FS as, d0ln|PQ0¯|, is added to the original source nodes in the traditional MFS, and the point charge d0(=q) and the source node Q0 are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node Q* is located inside S but near Γ, both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function qln|PQ*¯| as the source node Q* is located near Γ.

A novel localized collocation solver based on Trefftz basis for potential-based inverse electromyography

This paper introduces a novel localized collocation Trefftz method (LCTM) for potential-based inverse electromyography (PIE). PIE is a noninvasive technique to calculate the internal electrical potentials from measured body surface electromyographic data, which can be considered as an inverse Cauchy problem with potential equation. In the proposed LCTM, the electrical potential at every node is expressed as a linear combination of 3D Trefftz basis in each stencil support, and the sparse linear system is yield by satisfying governing equation at interior nodes and boundary conditions at boundary nodes. The proposed LCTM inherits the properties of easy-to-use and meshless from the collocation Trefftz method (CTM), and mitigates the ill-conditioning resultant matrix encountered in the CTM. Numerical efficiency of the proposed method is investigated in comparison with the CTM and experimental data.

On Solving Modified Helmholtz Equation in Layered Materials Using the Multiple Source Meshfree Approach

This paper presents a study for solving the modified Helmholtz equation in layered materials using the multiple source meshfree approach (MSMA). The key idea of the MSMA starts with the method of fundamental solutions (MFS) as well as the collocation Trefftz method (CTM). The multiple source collocation scheme in the MSMA stems from the MFS and the basis functions are formulated using the CTM. The solution of the modified Helmholtz equation is therefore approximated by the superposition theorem using particular nonsingular functions by means of multiple sources located within the domain. To deal with the two-dimensional modified Helmholtz equation in layered materials, the domain decomposition method was adopted. Numerical examples were carried out to validate the method. The results illustrate that the MSMA is relatively simple because it avoids a complicated procedure for finding the appropriate position of the sources. Additionally, the MSMA for solving the modified Helmholtz equation is advantageous because the source points can be collocated on or within the domain boundary and the results are not sensitive to the location of source points. Finally, compared with other methods, highly accurate solutions can be obtained using the proposed method.

A spacetime collocation Trefftz method for solving the inverse heat conduction problem

In this article, a novel spacetime collocation Trefftz method for solving the inverse heat conduction problem is presented. This pioneering work is based on the spacetime collocation Trefftz method; the method operates by collocating the boundary points in the spacetime coordinate system. In the spacetime domain, the initial and boundary conditions are both regarded as boundary conditions on the spacetime domain boundary. We may therefore rewrite an initial value problem (such as a heat conduction problem) as a boundary value problem. Hence, the spacetime collocation Trefftz method is adopted to solve the inverse heat conduction problem by approximating numerical solutions using Trefftz base functions satisfying the governing equation. The validity of the proposed method is established for a number of test problems. We compared the accuracy of the proposed method with that of the Trefftz method based on exponential basis functions. Results demonstrate that the proposed method obtains highly accurate numerical solutions and that the boundary data on the inaccessible boundary can be recovered even if the accessible data are specified at only one-fourth of the overall spacetime boundary.

On Solving Two-Dimensional Inverse Heat Conduction Problems Using the Multiple Source Meshless Method

In this article, a newly developed multiple-source meshless method (MSMM) capable of solving inverse heat conduction problems in two dimensions is presented. Evolved from the collocation Trefftz method (CTM), the MSMM approximates the solution by using many source points through the addition theorem such that the ill-posedness is greatly reduced. The MSMM has the same superiorities as the CTM, such as the boundary discretization only, and is advantageous for solving inverse problems. Several numerical examples are conducted to validate the accuracy of solving inverse heat conduction problems using boundary conditions with different levels of noise. Moreover, the domain decomposition method is adopted for problems in the doubly-connected domain. The results demonstrate that the proposed method may recover the unknown data with remarkably high accuracy, even though the over-specified boundary data are assigned a portion that is less than 1/10 of the overall domain boundary.

On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method

In this article, a solution to nonlinear moving boundary problems in heterogeneous geological media using the meshless method is proposed. The free surface flow is a moving boundary problem governed by Laplace equation but has nonlinear boundary conditions. We adopt the collocation Trefftz method (CTM) to approximate the solution using Trefftz base functions, satisfying the Laplace equation. An iterative scheme in conjunction with the CTM for finding the phreatic line with over–specified nonlinear moving boundary conditions is developed. To deal with flow in the layered heterogeneous soil, the domain decomposition method is used so that the hydraulic conductivity in each subdomain can be different. The method proposed in this study is verified by several numerical examples. The results indicate the advantages of the collocation meshless method such as high accuracy and that only the surface of the problem domain needs to be discretized. Moreover, it is advantageous for solving nonlinear moving boundary problems with heterogeneity with extreme contrasts in the permeability coefficient.

An efficient boundary collocation scheme for transient thermal analysis in large-size-ratio functionally graded materials under heat source load

This paper presents a boundary collocation scheme for transient thermal analysis in large-size-ratio functionally graded materials (FGMs) with heat source load. In the proposed scheme, Laplace transformation and the numerical inverse Laplace transformation (NILT) are implemented to avoid the troublesome time-stepping effect on numerical efficiency. The collocation Trefftz method (CTM) coupled with composite multiple reciprocity method is used to obtain the high accurate results in the solution of nonhomogeneous problems in Laplace-space domain. The extended precision arithmetic is introduced to overcome the ill-posed issues generated from the CTM simulation, the NILT process and the large-size-ratio FGM. Heuristic error analysis and numerical investigation are presented to demonstrate the effectiveness of the proposed scheme for transient thermal analysis. Several benchmark examples are considered under large-size-ratio FGMs with some specific spatial variations (quadratic, exponential and trigonometric functions). The proposed scheme is validated in comparison with known analytical solutions and COMSOL simulation.

A Novel Boundary-Type Meshless Method for Solving the Modified Helmholtz Equation

This paper presents a novel boundary-type meshless method for solving the two-dimensional modified Helmholtz equation in multiply connected regions. Numerical approximation is obtained by the superposition principle of the non-singular basis functions satisfied the governing equation. The advantage of the proposed method is that the locations of the source points are not sensitive to the results. The novel concept may resolve the major issue for the method of fundamental solutions (MFS). In contrast to the collocation Trefftz method (CTM), the Trefftz order of the non-singular basis functions can be reduced since the multiple source points are adopted. To solve the physical problems in a doubly and multiply connected regions, the boundary-type meshless method with the incorporation of the domain decomposition method (DDM) is proposed. The numerical examples for the validation are conducted including simply, doubly, and multiply connected regions. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient for modeling modified Helmholtz equation. The results also reveal that it resolves the restrictions of the MFS and the CTM for the application in the engineering problems.

A Novel Hybrid Boundary-Type Meshless Method for Solving Heat Conduction Problems in Layered Materials

In this article, we propose a novel meshless method for solving two-dimensional stationary heat conduction problems in layered materials. The proposed method is a recently developed boundary-type meshless method which combines the collocation scheme from the method of fundamental solutions (MFS) with the collocation Trefftz method (CTM) to improve the applicability of the method for solving boundary value problems. Particular non-singular basis functions from cylindrical harmonics are adopted in which the numerical approximation is based on the superposition principle using the non-singular basis functions expressed in terms of many source points. For the modeling of multi-layer composite materials, we adopted the domain decomposition method (DDM), which splits the domain into smaller subdomains. The continuity of the flux and the temperature has to be satisfied at the interface of subdomains for the problem. The validity of the proposed method is investigated for several test problems. Numerical applications were also carried out. Comparison of the proposed method with other meshless methods showed that it is highly accurate and computationally efficient for modeling heat conduction problems, especially in heterogeneous multi-layer composite materials.

On modeling subsurface flow using a novel hybrid Trefftz–MFS method

In this paper, a novel hybrid Trefftz–MFS method for modeling subsurface flow problems is presented. The proposed method constructs its nonsingular basis function as a series using T functions from the Trefftz method instead of using the singular solution in the method of fundamental solutions (MFS). Numerical solutions are approximated by superpositioning basis functions that are expressed in terms of many source points. Because of the use of the nonsingular basis function, the position of the source point in the proposed method is not sensitive to the results; thus, it resolves a major issue in the MFS for finding a satisfactory location for the source point. Additionally, the order of the nonsingular basis function can be reduced, because the addition theorem is used for approximating the solution for many source points. Consequently, the ill-posedness resulting from the adoption of higher order terms for the solution with only one source point in the collocation Trefftz method (CTM) is mitigated. The proposed method is validated for several test problems. Application examples are also performed. The results reveal that this novel method provides a promising solution that combines the benefits of the CTM and the MFS, such as the boundary collocation only and high accuracy. Moreover, the limitations to the practical application of the CTM and the MFS can be overcome using the proposed method.

A Novel Boundary-Type Meshless Method for Modeling Geofluid Flow in Heterogeneous Geological Media

A novel boundary-type meshless method for modeling geofluid flow in heterogeneous geological media was developed. The numerical solutions of geofluid flow are approximated by a set of particular solutions of the subsurface flow equation which are expressed in terms of sources located outside the domain of the problem. This pioneering study is based on the collocation Trefftz method and provides a promising solution which integrates the T-Trefftz method and F-Trefftz method. To deal with the subsurface flow problems of heterogeneous geological media, the domain decomposition method was adopted so that flux conservation and the continuity of pressure potential at the interface between two consecutive layers can be considered in the numerical model. The validity of the model is established for a number of test problems. Application examples of subsurface flow problems with free surface in homogenous and layered heterogeneous geological media were also carried out. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient. The results also reveal that it has great numerical stability for solving subsurface flow with nonlinear free surface in layered heterogeneous geological media even with large contrasts in the hydraulic conductivity.

Transient Modeling of Flow in Unsaturated Soils Using a Novel Collocation Meshless Method

In this paper, a novel meshless method for the transient modeling of subsurface flow in unsaturated soils was developed. A linearization process for the nonlinear Richards equation using the Gardner exponential model to analyze the transient flow in the unsaturated zone was adopted. For the transient modeling, we proposed a pioneering work using the collocation Trefftz method and utilized the coordinate system in Minkowski spacetime instead of that in the original Euclidean space. The initial value problem for transient modeling of subsurface flow in unsaturated soils can then be transformed into the inverse boundary value problem. A numerical solution obtained in the spacetime coordinate system was approximated by superpositioning Trefftz basis functions satisfying the governing equation for boundary collocation points on partial problem domain boundary in the spacetime coordinate system. As a result, the transient problems can be solved without using the traditional time-marching scheme. The validity of the proposed method is established for several test problems. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient. The results also reveal that it has great numerical stability for the transient modeling of subsurface flow in unsaturated soils.

Numerical solutions of elliptic partial differential equations using Chebyshev polynomials

We present a simple and effective Chebyshev polynomial scheme (CPS) combined with the method of fundamental solutions (MFS) and the equilibrated collocation Trefftz method for the numerical solutions of inhomogeneous elliptic partial differential equations (PDEs). In this paper, CPS is applied in a two-step approach. First, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then the resulting homogeneous equation is solved by boundary type methods including the MFS and the equilibrated collocation Trefftz method. Numerical results for problems on various irregular domains show that our proposed scheme is highly accurate and efficient.