scholarly journals Solving Transient Groundwater Inverse Problems Using Space–Time Collocation Trefftz Method

Water ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 3580
Author(s):  
Cheng-Yu Ku ◽  
Li-Dan Hong ◽  
Chih-Yu Liu
Keyword(s):  
Inverse Problems ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Separation Of Variables ◽  
Time Integration ◽  
Meshfree Method ◽  
Space Time ◽  
Basis Functions ◽  
Trefftz Method ◽  
Dimensional Boundary

This paper presents a space–time meshfree method for solving transient inverse problems in subsurface flow. Based on the transient groundwater equation, we derived the Trefftz basis functions utilizing the method of separation of variables. Due to the basis functions completely satisfying the equation to be solved, collocation points are placed on the space–time boundaries. Numerical solutions are approximated based on the superposition theorem. Accordingly, the initial and boundary conditions are both regarded as space–time boundary conditions. Forward and inverse examples are conducted to validate the proposed approach. Emphasis is placed on the two-dimensional boundary detection problem in which the nonlinearity is solved using the fictitious time integration method. Results demonstrate that approximations with high accuracy are acquired in which the boundary data on the absent boundary may be efficiently recovered even when inaccessible partial data are provided.

Hybrid Method of Engineering Analysis: Combining Meshfree Method with Distance Fields and Collocation Technique

2011 ◽  
Vol 11 (3) ◽  
Author(s):  
Igor Tsukanov ◽  
Sudhir R. Posireddy
Keyword(s):  
Boundary Conditions ◽  
Unstructured Grids ◽  
Numerical Technique ◽  
Meshfree Method ◽  
Basis Functions ◽  
Benchmark Problems ◽  
Hybrid Technique ◽  
Engineering Analysis ◽  
Distance Fields ◽  
Collocation Technique

This paper describes a numerical technique for solving engineering analysis problems that combine radial basis functions and collocation technique with meshfree method with distance fields, also known as solution structure method. The proposed hybrid technique enables exact treatment of all prescribed boundary conditions at every point on the geometric boundary and can be efficiently implemented for both structured and unstructured grids of basis functions. Ability to use unstructured grids empowers the meshfree method with distance fields with higher level of geometric flexibility. By providing exact treatment of the boundary conditions, the new approach makes it possible to exclude boundary conditions from the collocation equations. This reduces the size of the algebraic system, which results in faster solutions. At the same time, the boundary collocation points can be used to enforce the governing equation of the problem, which enhances the solution’s accuracy. Application of the proposed method to solution of heat transfer problems is illustrated on a number of benchmark problems. Modeling results are compared with those obtained by the traditional collocation technique and meshfree method with distance fields.

Hybrid Method of Engineering Analysis: Combining Meshfree Method With Distance Fields and Collocation Technique

2009 ◽  
Author(s):  
Igor Tsukanov ◽  
Sudhir R. Posireddy
Keyword(s):  
Boundary Conditions ◽  
Numerical Technique ◽  
Geometric Model ◽  
Meshfree Method ◽  
Basis Functions ◽  
Engineering Analysis ◽  
Alternative Analysis ◽  
Distance Fields ◽  
Analysis Methods ◽  
Collocation Technique

Most of the modern engineering analysis methods (Finite Element, Finite Difference, Finite Volume, etc.) rely on space discretizations of the underlying geometric model. Such spatial meshes have to conform to the geometric model in order to approximate boundary conditions, construct basis functions with good local properties as well as perform numerical integration and visualization of the modeling results. Despite recent advances in automatic mesh generation, spatial meshing still remains difficult problem which often requires geometry simplification and feature removal. Conforming spatial mesh also restricts motions and variations of the geometry and breaks design-analysis cycle. In order to overcome difficulties and restrictions of the mesh-based methods, the alternative analysis methods have been proposed. We present a numerical technique for solving engineering analysis problems that combines meshfree method with distance fields, radial basis functions and collocation technique. The proposed approach enhances the collocation method with exact treatment of boundary conditions at all boundary points. It makes it possible to exclude boundary conditions from the collocation equations. This reduces the size of the algebraic system which results in faster solutions. On another hand, the boundary collocation points can be used to enforce the governing equation of the problem which enhances the solutions accuracy. Ability to use unstructured grids empowers the meshfree method with distance fields with higher level of geometric flexibility. In our presentation we demonstrate comparisons of the numerical results given by the combined approach with results delivered by the traditional collocation technique and meshfree method with distance fields.

scholarly journals A Spacetime Meshless Method for Modeling Subsurface Flow with a Transient Moving Boundary

Water ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 2595 ◽  
Author(s):  
Cheng-Yu Ku ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Weichung Yeih ◽  
Chia-Ming Fan
Keyword(s):  
Diffusion Equation ◽  
Meshless Method ◽  
Moving Boundary ◽  
Iterative Scheme ◽  
Separation Of Variables ◽  
Subsurface Flow ◽  
Basis Functions ◽  
Method Of Fundamental Solutions ◽  
Trefftz Method ◽  
Flow Problems

In this paper, a spacetime meshless method utilizing Trefftz functions for modeling subsurface flow problems with a transient moving boundary is proposed. The subsurface flow problem with a transient moving boundary is governed by the two-dimensional diffusion equation, where the position of the moving boundary is previously unknown. We solve the subsurface flow problems based on the Trefftz method, in which the Trefftz basis functions are obtained from the general solutions using the separation of variables. The solutions of the governing equation are then approximated numerically by the superposition theorem using the basis functions, which match the data at the spacetime boundary collocation points. Because the proposed basis functions fully satisfy the diffusion equation, arbitrary nodes are collocated only on the spacetime boundaries for the discretization of the domain. The iterative scheme has to be used for solving the moving boundaries because the transient moving boundary problems exhibit nonlinear characteristics. Numerical examples, including harmonic and non-harmonic boundary conditions, are carried out to validate the method. Results illustrate that our method may acquire field solutions with high accuracy. It is also found that the method is advantageous for solving inverse problems as well. Finally, comparing with those obtained from the method of fundamental solutions, we may obtain the accurate location of the nonlinear moving boundary for transient problems using the spacetime meshless method with the iterative scheme.

A Meshfree Computational Approach Based on Multiple-Scale Pascal Polynomials for Numerical Solution of a 2D Elliptic Problem with Nonlocal Boundary Conditions

2019 ◽  
Vol 17 (10) ◽  
pp. 1950080 ◽  
Author(s):  
Ömer Oruç
Keyword(s):  
Boundary Conditions ◽  
Elliptic Problem ◽  
Numerical Solutions ◽  
Nonlocal Boundary ◽  
Meshfree Method ◽  
Multiple Scale ◽  
Nonlocal Boundary Conditions ◽  
Condition Numbers ◽  
Test Problems ◽  
Numerical Integrations

A two-dimensional (2D) elliptic problem with nonlocal boundary conditions on both regular and irregular domains is solved numerically by Pascal polynomial basis unified with multiple-scale technique. Very accurate numerical solutions and quite reasonable condition numbers are obtained with the proposed method which is also a truly meshfree method since difficult meshing processes or numerical integrations over domains are not needed for considered problems. Four test problems are solved to show the accuracy and efficiency of the proposed method. Also stability of the method is studied against large noise effect.

scholarly journals Use of radial basis functions for meshless numerical solutions applied to financial engineering barrier options

2009 ◽  
Vol 29 (2) ◽  
pp. 419-437 ◽  
Author(s):  
Gisele Tessari Santos ◽  
Maurício Cardoso de Souza ◽  
Mauri Fortes
Keyword(s):  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Analytical Solutions ◽  
Financial Engineering ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Barrier Options ◽  
Black Scholes ◽  
Non Linear ◽  
Radial Basis

A large number of financial engineering problems involve non-linear equations with non-linear or time-dependent boundary conditions. Despite available analytical solutions, many classical and modified forms of the well-known Black-Scholes (BS) equation require fast and accurate numerical solutions. This work introduces the radial basis function (RBF) method as applied to the solution of the BS equation with non-linear boundary conditions, related to path-dependent barrier options. Furthermore, the diffusional method for solving advective-diffusive equations is explored as to its effectiveness to solve BS equations. Cubic and Thin-Plate Spline (TPS) radial basis functions were employed and evaluated as to their effectiveness to solve barrier option problems. The numerical results, when compared against analytical solutions, allow affirming that the RBF method is very accurate and easy to be implemented. When the RBF method is applied, the diffusional method leads to the same results as those obtained from the classical formulation of Black-Scholes equation.

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

2019 ◽  
Vol 11 (7) ◽  
pp. 168781401986127 ◽  
Author(s):  
Cheng-Yu Ku ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Wei-Po Huang ◽  
Yan Su
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Heat Conduction Problem ◽  
Domain Boundary ◽  
Inverse Heat Conduction Problem ◽  
Test Problems ◽  
Inverse Heat Conduction ◽  
Trefftz Method ◽  
Base Functions

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.

scholarly journals A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 113 ◽  
Author(s):  
Muaz Seydaoğlu
Keyword(s):  
Lie Group ◽  
Kinematic Viscosity ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Time Integration ◽  
Group Scheme ◽  
Basis Functions ◽  
One Dimensional ◽  
Radial Basis ◽  
The One

An efficient technique is proposed to solve the one-dimensional Burgers’ equation based on multiquadric radial basis function (MQ-RBF) for space approximation and a Lie-Group scheme for time integration. The comparisons of the numerical results obtained for different values of kinematic viscosity are made with the exact solutions and the reported results to demonstrate the efficiency and accuracy of the algorithm. It is shown that the numerical solutions concur with existing results and the proposed algorithm is efficient and can be easily implemented.

scholarly journals An Improved Version of Residual Power Series Method for Space-Time Fractional Problems

2022 ◽  
Vol 2022 ◽  
pp. 1-9
Author(s):  
Mine Aylin Bayrak ◽  
Ali Demir ◽  
Ebru Ozbilge
Keyword(s):  
Power Series ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Space Time ◽  
Dirichlet Boundary Conditions ◽  
Power Series Method ◽  
Residual Power Series ◽  
Residual Power

The task of present research is to establish an enhanced version of residual power series (RPS) technique for the approximate solutions of linear and nonlinear space-time fractional problems with Dirichlet boundary conditions by introducing new parameter λ . The parameter λ allows us to establish the best numerical solutions for space-time fractional differential equations (STFDE). Since each problem has different Dirichlet boundary conditions, the best choice of the parameter λ depends on the problem. This is the major contribution of this research. The illustrated examples also show that the best approximate solutions of various problems are constructed for distinct values of parameter λ . Moreover, the efficiency and reliability of this technique are verified by the numerical examples.

Identification of the string boundary conditions using natural frequencies

2016 ◽  
Vol 11 (1) ◽  
pp. 38-52
Author(s):  
I.M. Utyashev ◽  
A.M. Akhtyamov
Keyword(s):  
Inverse Problems ◽  
Power Series ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Boundary Value ◽  
External Environment ◽  
Direct And Inverse Problems ◽  
Symmetric Potential ◽  
Modified Method

The paper discusses direct and inverse problems of oscillations of the string taking into account symmetrical characteristics of the external environment. In particular, we propose a modified method of finding natural frequencies using power series, and also the problem of identification of the boundary conditions type and parameters for the boundary value problem describing the vibrations of a string is solved. It is shown that to identify the form and parameters of the boundary conditions the two natural frequencies is enough in the case of a symmetric potential q(x). The estimation of the convergence of the proposed methods is done.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

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