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

Ku, Cheng-Yu; Liu, Chih-Yu; Xiao, Jing-En; Huang, Wei-Po; Su, Yan

doi:10.1177/1687814019861271

Cited by 6 publications

(3 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The governing equation, as written as Equation ( 1), is considered. The boundary values on the right and left boundaries are described as Equations (19) and (20). The boundary data at final time is:…”

Section: The Backward Heat Conduction Problem (Bhcp)mentioning

confidence: 99%

“…As a result, the time-marching methods for solving heat conduction problems may be computationally expensive. Recently, several numerical approaches based on the space-time scheme have been proposed [16][17][18][19][20]. Tezduyar et al proposed the space-time finite element method (FEM) for the computation of fluid-structure interaction problems [17].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Solving Backward Heat Conduction Problems Using a Novel Space–Time Radial Polynomial Basis Function Collocation Method

Liu

Xiao

et al. 2020

Applied Sciences

Self Cite

View full text Add to dashboard Cite

In this article, a novel meshless method using space–time radial polynomial basis function (SRPBF) for solving backward heat conduction problems is proposed. The SRPBF is constructed by incorporating time-dependent exponential function into the radial polynomial basis function. Different from the conventional radial basis function (RBF) collocation method that applies the RBF at each center point coinciding with the inner point, an innovative source collocation scheme using the sources instead of the centers is first developed for the proposed method. The randomly unstructured source, boundary, and inner points are collocated in the space–time domain, where both boundary as well as initial data may be regarded as space–time boundary conditions. The backward heat conduction problem is converted into an inverse boundary value problem such that the conventional time–marching scheme is not required. Because the SRPBF is infinitely differentiable and the corresponding derivative is a nonsingular and smooth function, solutions can be approximated by applying the SRPBF without the shape parameter. Numerical examples including the direct and backward heat conduction problems are conducted. Results show that more accurate numerical solutions than those of the conventional methods are obtained. Additionally, it is found that the error does not propagate with time such that absent temperature on the inaccessible boundaries can be recovered with high accuracy.

show abstract

Section: The Backward Heat Conduction Problem (Bhcp)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solving Backward Heat Conduction Problems Using a Novel Space–Time Radial Polynomial Basis Function Collocation Method

Liu

Xiao

et al. 2020

Applied Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

“…For such problems, the use of the meshfree methods to acquire approximate solutions is advantageous. Several meshfree methods utilizing approximation functions, such as moving least squares, reproducing kernel collocation method, Trefftz method, and element-free Galerkin method [5,6], have been widely used for solving engineering problems. The complexities involved in the solution of the governing equations require advanced mathematical approaches, such as the radial basis function collocation method (RBFCM) [7,8].…”

Section: Introductionmentioning

confidence: 99%

Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

Liu

Xiao

et al. 2020

Symmetry

Self Cite

View full text Add to dashboard Cite

In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.

show abstract

Application of the image‐well method for transient borehole thermal energy storage systems with complex boundaries

Lin,

Rau,

Kurylyk

2024

Heat Trans

View full text Add to dashboard Cite

This study introduces a new analytical framework that employs the image‐well method to simulate the spatial and temporal temperature distribution in vertical borehole thermal energy storage (BTES) systems. The model accommodates complex boundary shapes and conditions, including insulation, convection, and constant temperature, without requiring iterative solutions at each time step. The model's accuracy and utility are demonstrated through an application to a borehole heat exchanger cluster arranged in an octagonal shape with insulating boundaries, based on a BTES site in Drake Landing (Canada). Model predictions are validated against a finite element model, showing a root‐mean‐square error of 0.012°C. A global sensitivity analysis highlights the influence of thermal parameters on system performance, identifying the heat flux of the borehole heat exchanger as the most sensitive parameter. Overall, this approach combines the advantages of analytical and numerical techniques to provide a clear and efficient tool for evaluating BTES systems, offering significant potential for advancing sustainable energy solutions.

show abstract

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

Cited by 6 publications

References 25 publications

Solving Backward Heat Conduction Problems Using a Novel Space–Time Radial Polynomial Basis Function Collocation Method

Solving Backward Heat Conduction Problems Using a Novel Space–Time Radial Polynomial Basis Function Collocation Method

Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

Application of the image‐well method for transient borehole thermal energy storage systems with complex boundaries

Contact Info

Product

Resources

About