Matched Interface And Boundary Research Articles

A Time-Domain Wavenumber Integration Model for Underwater Acoustics Based on the High-Order Finite Difference Method

Simulating the acoustic field excited by pulse sound sources holds significant practical value in computational ocean acoustics. Two primary methods exist for modeling underwater acoustic propagation in the time domain: the Fourier synthesis technique based on frequency decomposition and the time-domain underwater acoustic propagation model (TD-UAPM). TD-UAPMs solve the wave equation in the time domain without requiring frequency decomposition, providing a more intuitive explanation of the physical process of sound energy propagation over time. However, time-stepping numerical methods can accumulate numerical errors, making it crucial to improve the algorithm’s accuracy for TD-UAPMs. Herein, the time-domain wavenumber integration model SPARC was improved by replacing the second-order finite element method (FEM) with the high-order accuracy finite difference method (FDM). Furthermore, the matched interface and boundary (MIB) method was used to process the seabed more accurately. The improved model was validated using three classic underwater acoustic benchmarks. By comparing the acoustic solutions obtained using the FDM and the FEM, it is evident that the improved model requires fewer grid points while maintaining the same level of accuracy, leading to lower computational costs and faster processing compared to the original model.

A new MIB-based time integration method for transient heat conduction analysis of discrete and continuous systems

By incorporating the matched interface and boundary (MIB) method into time integration computation, a new MIB-based time integration (MIBTI) approach is proposed for solving transient heat conduction problems in this work. Indeed, this is a truly effective time-domain MIB method that can directly discretize space and time derivatives, and the transient heat conduction problem (an initial-value problem) is transformed into a pseudo-space boundary-value problem. The initial and final time points are regarded as boundary points in the transformed space domain. Two algorithms are developed to deal with the final time point. Unlike conventional recursive schemes for time integration, all unknown variables can be determined by solving the transformed boundary-value problem in the present approach. Various discrete and continuous systems without and with internal heat sources under different boundary conditions are considered to examine the working performance of the proposed method. The applicability of the time stability criterion in the conventional recursive schemes to the proposed method is also discussed. It is found that excellent numerical stability and convergence can be achieved by using the present MIBTI method for solving transient heat conduction problems.

Comparison of three matched interface and boundary (MIB) schemes for solving the nonlinear Poisson-Boltzmann equation

Comparison of three matched interface and boundary (MIB) schemes for solving the nonlinear Poisson-Boltzmann equation

A FFT accelerated fourth order finite difference method for solving three-dimensional elliptic interface problems

In this paper, a fourth order augmented matched interface and boundary (AMIB) method is proposed for solving a three-dimensional elliptic interface problem which involves a smooth material interface inside a cuboid domain. On the boundary of the cuboid domain, the fourth order AMIB method can handle different types of boundary conditions, including Dirichlet, Neumann, Robin and their mix combinations in fictitious value generation. Moreover, zero-padding solutions are introduced so that the fast Fourier transform (FFT) algorithm is still valid near the boundary. In dealing with the interior interface, a fourth order ray-casting matched interface and boundary (MIB) scheme is proposed, which enforces the jump conditions along the normal direction for calculating fictitious values. Comparing with the existing MIB scheme, the ray-casting scheme naturally bypasses the corner issue and becomes more robust in handling complex geometry. Based on fictitious values generated near interface and boundary, the fourth order central difference can be corrected at various irregular points including corner points, by introducing Cartesian derivative jumps as auxiliary variables. This gives rise to an enlarged linear system, which can be efficiently solved by the Schur complement procedure together with the FFT inversion of the discrete Laplacian. Extensive numerical experiments have been carried to test the proposed ray-casting AMIB method for numerical accuracy, efficiency, and robustness in corner treatment. The numerical results demonstrate that the ray-casting AMIB scheme not only maintains a fourth order of accuracy in treating various interfaces and boundaries for both solutions and solution gradients, but also attains an overall efficiency on the order of O(n3log⁡n) for a n×n×n uniform grid.

Free Vibration Analysis of Rectangular Thin Plates Resting on Nonhomogeneous Elastic Foundations by Using Matched Interface and Boundary Algorithms and their Multi-Domain Formulations

Matched interface and boundary (MIB) algorithms and their multi-domain formulations are developed to solve free vibration problems of thin plates resting on unidirectionally and bidirectionally nonhomogeneous elastic foundations. Winkler and Pasternak types of elastic foundations are considered. Three types of MIB algorithms are introduced to handle unidirectional interfaces caused by nonhomogeneous foundations. On the basis of in-depth understanding of MIB method and Kirchhoff plate theory, new multi-domain MIB (MD-MIB) algorithms are proposed to handle bidirectional interfaces caused by nonhomogeneous foundations that cannot be handled by using the existing MIB method. Three types of MD-MIB algorithms are developed by using three domain decomposition strategies. A number of examples are chosen to validate the applicabilities of different MIB and MD-MIB algorithms to the title problems. Numerical results obtained by using MIB and MD-MIB algorithms are compared with the existing solutions or finite element solutions. Numerical studies show that MIB and MD-MIB algorithms are efficient approaches to deal with nonhomogeneous elastic foundations.

A series of multi-domain matched interface and boundary algorithms for dynamic and static responses of annular sectorial plates

A series of multi-domain matched interface and boundary (MDMIB) algorithms are proposed to analyze the free vibration and bending of annular sectorial thin plates. Through an in-depth understanding of the matched interface and boundary (MIB) method, six types of MDMIB algorithms are developed herein by using various domain decomposition techniques for handling bi-directional crossing interfaces due to non-homogeneous elastic foundations and discontinuous loads, which cannot be treated by the conventional MIB method. The rationale of the MDMIB method is to divide the entire computational domain into multiple subdomains that can be connected by using the interconnecting conditions between crossing interfaces and mesh lines. Annular sectorial plates, having non-uniform boundary conditions, partially supported elastic foundations, and patch loads, are all considered. Numerous examples are chosen to examine the computational performance of the MDMIB algorithms. For verification, the present results are checked against existing and finite element solutions. It is shown that the proposed MDMIB algorithms are highly effective to deal with the bi-directional interface problems.

Matched Interface and Boundary Method for the Helmholtz Equation with a Discretization by 4th-order 17-point FD Scheme

Abstract We generalize the MIB (matched interface and boundary) method for solving the Helmholtz equation with a rotated 17-point 4th-order FD(finite difference) scheme, which is non-compact, and has to use the points outside of the computational region. An optimal combination of difference schemes are employed to discretize the Helmholtz equation both on the standard coordinate system and the 45° rotated one. The combination weights are obtained via minimizing the numerical dispersion. Numerical experiments show that the rotated 17-point FD scheme with the MIB method is 4th-order in accuracy, and is effective in supressing numerical dispersion.

A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains

For elliptic boundary value problems (BVPs) involving irregular domains and Robin boundary condition, no numerical method is known to deliver a fourth order convergence and O(Nlog⁡N) efficiency, where N stands for the total degree-of-freedom of the system. Based on the matched interface and boundary (MIB) and fast Fourier transform (FFT) schemes, a new finite difference method is introduced for such problems, which involves two main components. First, a ray-casting MIB scheme is proposed to handle different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mix combinations. By enclosing the concerned irregular domain by a large enough cubic domain, the ray-casting MIB scheme generates necessary fictitious values outside the irregular domain by imposing boundary conditions along the normal direction of the boundary, so that a high order central difference discretization of the Laplacian can be formed. Second, an augmented MIB formulation is built, in which Cartesian derivative jumps are reconstructed on the boundary as auxiliary variables. By treating such variables as unknowns, the discrete Laplacian can be efficiently inverted by the FFT algorithm, in the Schur complement solution of the augmented system. The accuracy and efficiency of the proposed augmented MIB method are numerically examined by considering various elliptic BVPs in two and three dimensions. Numerical results indicate that the new algorithm not only achieves a fourth order of accuracy in treating irregular domains and complex boundary conditions, but also maintains the FFT efficiency.

A multigrid based finite difference method for solving parabolic interface problem

<p style='text-indent:20px;'>In this paper, a new Cartesian grid finite difference method is introduced to solve two-dimensional parabolic interface problems with second order accuracy achieved in both temporal and spatial discretization. Corrected central difference and the Matched Interface and Boundary (MIB) method are adopted to restore second order spatial accuracy across the interface, while the standard Crank-Nicolson scheme is employed for the implicit time stepping. In the proposed augmented MIB (AMIB) method, an augmented system is formulated with auxiliary variables introduced so that the central difference discretization of the Laplacian could be disassociated with the interface corrections. A simple geometric multigrid method is constructed to efficiently invert the discrete Laplacian in the Schur complement solution of the augmented system. This leads a significant improvement in computational efficiency in comparing with the original MIB method. Being free of a stability constraint, the implicit AMIB method could be asymptotically faster than explicit schemes. Extensive numerical results are carried out to validate the accuracy, efficiency, and stability of the proposed AMIB method.</p>

Comparisons of matched interface and boundary (MIB) method and its interpolation formulation for free vibration analysis of stepped beams and plates

This work conducts a comparison study of matched interface and boundary (MIB) method and its interpolation formulation for free vibration analysis of stepped beams and plates. Two types of schemes are used to describe the relations between interfaces and grid points in one-dimensional and two-dimensional domains. In the first scheme, the interfaces are located on grid points, while, in the second scheme, the interfaces lie between grid points. Detailed procedures of MIB and the interpolation formulation of MIB (IMIB) are presented in these two schemes. Based on the in-depth understanding of MIB, a new algorithm is also developed to deal with cross derivatives of interface conditions of plates with interfaces lying between grid points to improve the accuracy. Various examples of stepped beams and plates are chosen to illustrate the performance of MIB and IMIB. This research work reveals that on the whole, MIB and IMIB are equivalent in the solutions of eigenvalues of beams and plates. However, some differences can also be observed for very large computational bandwidths. Some important conclusions are drawn at the end of this study.

Regularization methods for the Poisson-Boltzmann equation: Comparison and accuracy recovery

A significant challenge in numerical solution to the Poisson-Boltzmann equation is due to singular charge sources in terms of Dirac delta functions. To overcome this difficulty, several regularization methods have been developed, in which the potential function is decomposed into two or three parts so that the singular component can be analytically solved using the Green's function, while other components become bounded. However, it was observed in the literature that some regularization methods are significantly less accurate than the others for unclear reasons, even though they are analytically equivalent. To understand this discrepancy, the numerical performance of four popular regularization methods is investigated in this work by implementing them with the Matched Interface and Boundary (MIB) approach, which is a sophisticated finite difference method for treating elliptic interface problems with discontinuous coefficients. With all four methods showing second order convergence, accuracy reduction is numerically observed in two schemes. This paper provides numerical analysis and experiment to trace the source of such reduction, and links the error to the fact that the Laplacian of Green's function is dropped outside the protein domain. While this term is analytically vanishing, its numerical negligence introduces a discretization error. Formulating via a proper elliptic interface problem, an effective accuracy recovery technique is proposed so that all four methods yield the same high precision. With this study, all involved regularization schemes are better understood and well connected into a unified framework.

Development of Matched Interface and Boundary Method for Buckling Analysis of Plates with Various Interfaces

Buckling problems of plates with interfaces caused by step changes in thickness, internal line supports and line hinges under uni-axial and bi-axial in-plane compressive loads are solved by using the matched interface and boundary (MIB) method. In view of buckling problems of plates, new MIB algorithms and their interpolation formulations are developed to deal with various interfaces. A number of examples are taken to examine the accuracy and convergence of the present algorithms. Numerical results are compared with the existing solutions to validate the applicability of these algorithms to the title problems. Numerical analysis shows that on the whole, MIB and its interpolation formulation are equivalent, and they are highly accurate and efficient approaches for buckling analysis of plates with various interfaces.

A fourth order finite difference method for solving elliptic interface problems with the FFT acceleration

In this paper, a new Cartesian grid finite difference method is introduced based on the fourth order accurate matched interface and boundary (MIB) method and fast Fourier transform (FFT). The proposed augmented MIB method consists of two major parts for solving elliptic interface problems in two dimensions. For the interior part, in treating a smoothly curved interface, zeroth and first order jump conditions are enforced repeatedly by the MIB scheme to generate fictitious values near the interface. For the exterior part, two layers of zero-padding solutions are introduced beyond the original rectangular domain so that the FFT inversion becomes feasible. Different types of boundary conditions, including Dirichlet, Neumann, Robin and their mix combinations, can be imposed via the MIB scheme to generate fictitious values near boundaries. Based on fictitious values at both interfaces and boundaries, the augmented MIB method reconstructs Cartesian derivative jumps as auxiliary variables. Then, by treating such variables as unknowns, an enlarged linear system is obtained. In the Schur complement solution of such system, the FFT algorithm will not sense the solution discontinuities, so that the discrete Laplacian can be efficiently inverted. Therefore, the FFT-based augmented MIB not only achieves a fourth order of accuracy in dealing with interfaces and boundaries, but also produces an overall complexity of O(n2log⁡n) for a n×n uniform grid. Moreover, the augmented MIB scheme can provide fourth order accurate approximations to solution gradients and fluxes.

Alternating direction ghost-fluid methods for solving the heat equation with interfaces

This work presents two new alternating direction implicit (ADI) schemes for solving parabolic interface problems in two and three dimensions. First, the ghost fluid method (GFM) is adopted for the first time in the literature to treat interface jumps in the ADI framework, which results in symmetric and tridiagonal finite difference matrices in each ADI step. The proposed GFM-ADI scheme achieves a first order of accuracy in both space and time, as confirmed by numerical experiments involving complex interface shapes and spatial–temporal dependent jumps. The GFM-ADI scheme also maintains the ADI efficiency — the computational complexity for each time step scales as O(N) for a total degree of freedom N in higher dimensions. Second, a new matched interface and boundary (MIB) scheme is constructed, which downgrades the quadratic polynomial bases in the existing second order MIB to linear ones. Interestingly, the resulting MIB-ADI or mADI scheme produces the same finite difference matrices as the GFM-ADI scheme in all dimensions. Hence, the mADI scheme can be regarded as an improvement of the GFM, because it calculates tangential jumps which are omitted in the GFM. Consequently, the present mADI scheme is constantly more accurate than the GFM-ADI in all numerical examples, while keeping the same computational efficiency. Nevertheless, the mADI scheme is semi-implicit due to tangential jump approximations, while the GFM-ADI scheme is fully implicit without tangential corrections. Thus, the GFM-ADI scheme could be more stable than the mADI scheme when a huge contrast is presented in diffusion coefficients.

Matched Interface and Boundary Method for Free Vibration Analysis of Irregular Membranes

Matched interface and boundary (MIB) method is introduced for free vibration analysis of irregular membranes. Two distinct schemes-on-interface and off-interface schemes are used to deal with the topological relations between edges of irregular domains and the Cartesian mesh lines. Different geometric shapes such as triangle and quadrilateral are dealt with by using MIB procedures. A number of examples are chosen to demonstrate the accuracy and convergence of MIB method. Numerical results show that MIB method is an efficient and highly accurate approach to solve free vibration problems of irregular membranes. This study further extends the application of MIB.

A fourth-order 9-point finite difference method for the Helmholtz equation

In this paper, we develop a new fourth-order 9-point finite difference scheme for solving the Helmholtz equation. The central fourth-order difference scheme is used to discretize the second-order derivative, and the matched interface and boundary (MIB) method is employed to deal with the resulting boundary problem. For the discretization of the zero-order term, a weighted average method is designed by utilizing all of the 9 points, and the weight parameters are determined by minimizing the numerical dispersion. The new method is simple and efficient in suppressing the numerical dispersion. Finally, numerical examples are presented to illustrate the numerical convergence and effectiveness of the new scheme.

An augmented matched interface and boundary (MIB) method for solving elliptic interface problem

Abstract In this paper, a second order accurate augmented matched interface and boundary (MIB) is introduced for solving two-dimensional (2D) elliptic interface problems with piecewise constant coefficients. The augmented MIB seamlessly combines several key ingredients of the standard MIB, augmented immersed interface method (IIM), and explicit jump IIM, to produce a new fast interface algorithm. Based on the MIB, zeroth and first order jump conditions are enforced across an arbitrarily curved interface, which yields fictitious values on Cartesian nodes near the interface. By using such fictitious values, a simple procedure is proposed to reconstruct Cartesian derivative jumps as auxiliary variables and couple them with the jump-corrected Taylor series expansions, which allow us to restore the order of the central difference across the interface to two. Moreover, by using the Schur complement to disassociate the algebraic computation of auxiliary variables and function values, the discrete Laplacian can be efficiently inverted by using the fast Fourier transform (FFT). It is found in our numerical experiments that the iteration number in solving the auxiliary system weakly depends on the mesh size. As a consequence, the total computational cost of the augmented MIB is about O ( n 2 log n ) for a Cartesian grid with dimension n × n in 2D. Therefore, the augmented MIB outperforms the classical MIB in all cases by significantly reducing the CPU time, while keeping the same second order of accuracy in dealing with complicated interfaces.

A Comparison Study of HO-CFD and DSC-RSK with Small Computational Bandwidths for Solving Some Classes of Boundary-Value and Eigenvalue Problems

This paper conducts a comparison analysis of high order central finite difference (HO-CFD) method and discrete singular convolution-regularized Shannon kernel (DSC-RSK) scheme with small computational bandwidths for solving some classes of boundary-value and eigenvalue problems. Second-, fourth- and sixth-order partial differential equations are taken into account. New strategies to generate parameters [Formula: see text] in DSC-RSK are proposed to ensure minimum errors for each case, and the influence of parameters [Formula: see text] with more decimal places is analyzed. Apart from the existing matched interface and boundary (MIB) scheme, a new double-parameter MIB scheme is also proposed. The influence of small computational bandwidths is discussed in detail. Numerical results by using HO-CFD and DSC-RSK are presented and compared to illustrate the performance of both methods in small bandwidth limit. Some remarkable conclusions have been drawn at the end of this study.

Accurate p$K_a$ Computation Using Matched Interface and Boundary (MIB) Method Based Poisson-Boltzmann Solver

Accurate p$K_a$ Computation Using Matched Interface and Boundary (MIB) Method Based Poisson-Boltzmann Solver

A two-component Matched Interface and Boundary (MIB) regularization for charge singularity in implicit solvation

We present a new Matched Interface and Boundary (MIB) regularization method for treating charge singularity in solvated biomolecules whose electrostatics are described by the Poisson–Boltzmann (PB) equation. In a regularization method, by decomposing the potential function into two or three components, the singular component can be analytically represented by the Green's function, while other components possess a higher regularity. Our new regularization combines the efficiency of two-component schemes with the accuracy of the three-component schemes. Based on this regularization, a new MIB finite difference algorithm is developed for solving both linear and nonlinear PB equations, where the nonlinearity is handled by using the inexact-Newton's method. Compared with the existing MIB PB solver based on a three-component regularization, the present algorithm is simpler to implement by circumventing the work to solve a boundary value Poisson equation inside the molecular interface and to compute related interface jump conditions numerically. Moreover, the new MIB algorithm becomes computationally less expensive, while maintains the same second order accuracy. This is numerically verified by calculating the electrostatic potential and solvation energy on the Kirkwood sphere on which the analytical solutions are available and on a series of proteins with various sizes.