Irregular Computational Domains Research Articles

An efficient approach of meshless node placement in three-dimensional subsurface flow modeling

Meshless methods are valuable tools for real-time subsurface flow modeling and are particularly beneficial for adaptively adjusting node positions when incorporating supplementary measurements. However, an algorithm is required for automatically generating high-quality meshless nodes to adapt to irregularly shaped aquifers and arbitrarily distributed wells. This paper introduces a three-dimensional adaptive meshless node placement technique based on advancing front nodes and proposes a sigmoid exclusion circle to ensure adaptability to specific positions of interest. It is crucial for groundwater simulations to have specified computational points at wellbore locations. The generated meshless nodes were used as computational points in the generalized finite difference method. The node quality was assessed by comparing the numerical solutions to the analytical solution, achieving errors of the order of 10–12. The application scenarios of irregularly shaped aquifers demonstrated the high efficiency of the three-dimensional node placement in generating up to 10,000 nodes in <5 s. The effectiveness of the algorithm was verified using three-dimensional irregular computational domains. The algorithm could be easily adapted to two-dimensional problems, and an irregular boundary example is presented. The validation and application cases highlighted the versatility of the proposed method and established its potential for real-world hydrogeological applications.

Numerical study of the multi-dimensional Galilei invariant fractional advection–diffusion equation using direct mesh-less local Petrov–Galerkin method

This article presents a local mesh-less procedure for simulating the Galilei invariant fractional advection–diffusion (GI-FAD) equations in one, two, and three-dimensional spaces. The proposed method combines a second-order Crank–Nicolson scheme for time discretization and the second-order weighted and shifted Grünwald difference (WSGD) formula. This time discretization scheme ensures unconditional stability and convergence with an order of O(τ2). In the spatial domain, a mesh-less weak form is employed based on the direct mesh-less local Petrov–Galerkin (DMLPG) method. The DMLPG method employs the generalized moving least-square (GMLS) approximation in conjunction with the local weak form of the equation. By utilizing simple polynomials as shape functions in the GMLS approximation, the necessity for complex shape function construction in the MLS approximation is eliminated. To validate and demonstrate the effectiveness of the proposed algorithm, a variety of problems in one, two, and three dimensions are investigated on both regular and irregular computational domains. The numerical results obtained from these investigations confirm the accuracy and reliability of the developed approach in simulating GI-FAD equations.

Integrated radial basis function technique to simulate the nonlinear system of time fractional distributed-order diffusion equation with graded time-mesh discretization

The distributed-order fractional calculus (DOFC) is a generalization of the fractional calculus which its application can be found in viscoelasticity, transport processes and control theory. In the current paper, a system of time fractional distributed-order diffusion equation is investigated, numerically. In the first stage, the time derivative is approximated by a finite difference formulation. The integral terms are approximated by the numerical integration. Then, a semi-discrete scheme is constructed by this procedure. In the second stage, The stability and convergence of the time-discrete outline are analyzed by the energy method. In the third stage, the space derivative is discretized by the compact integrated radial basis function (CLIRBF) as a truly meshless method. Also, the numerical procedures are performed on regular and irregular computational domains. The numerical experiments verify the ability, efficiency and accuracy of the developed numerical formulation.

A local meshless method for solving multi-dimensional Galilei invariant fractional advection–diffusion equation

In this article, a local meshless technique is applied for numerical simulation of multi-dimensional Galilei invariant fractional advection–diffusion model on regular and irregular computational domains. In the suggested method, a second-order Crank–Nicolson scheme along with the second-order weighted and shifted Grünwald difference (WSGD) formula, is used to discretize the time derivatives of the model. This time-discretization scheme is unconditionally stable and convergent with order O(τ2). To approximate the spatial derivatives of this model, a local radial point interpolation technique is employed. Finally, to prove and demonstrate the validity of the proposed algorithm, various one, two and three-dimensional problems are investigated on regular and irregular computational domains.

Artificial neural network approximations of Cauchy inverse problem for linear PDEs

A novel artificial neural network method is proposed for solving Cauchy inverse problems. Using multiple-layers network as an approximation we present a non-mesh discretization to solve the problems. The existence and convergence are shown to establish the well-posedness of neural network approximations for the Cauchy inverse problems. Numerical results on 2D to 8D cases show that compared to finite element method, the neural network approach easier extends to high dimensional case. The stability and accuracy of the proposed network approach are investigated by the experiments with noisy boundary and irregular computational domain. Our studies conclude that the neural network method alleviates the influence of noise and it is observed that networks with wider and deeper hidden layers could lead to better approximation.

Numerical Study of Grain Growth in Laser Powder Bed Fusion Additive Manufacturing of Metals

Laser Powder Bed Fusion (LPBF) is an additive manufacturing method that manufactures high density and quality metal products. We present a coupled grain growth and heat transfer modeling technique to understand the materials microstructure evolution in metals during the cooling process of LPBF. The phase-field model is combined with a transient heat transfer equation to simulate the solidification and crystallization of the melt pool simultaneously. Specifically, the variable domain and driving force of the order parameters in the phase-field calculation are defined using current temperature distribution. Additionally, the latent heat generated by crystallization is introduced as a heat source to affect temperature evolution in the cooling process. The finite element method with a staggering strategy is employed to solve the coupled governing equations on an irregular computational domain. The computational framework is verified in a one-dimensional solidification problem by comparing the velocity of the fluid-solid interface. The two-way coupling solution of solidification and crystallization is studied in an example of LPBF of Aluminum alloys.

A trustable shape parameter in the kernel-based collocation method with application to pricing financial options

In this paper, we focus on the kernel-based solution of high-dimensional elliptic PDEs and propose an efficient algorithm to compute a trustable value of the shape parameter for a given positive definite kernel. The proposed strategy is based on the norm-minimal generalized interpolation problem coming from the optimality of radial basis function interpolation. Also, the effectiveness of the suggested algorithm is assessed by solving parabolic partial integro-differential equations arising in option pricing theory when the dynamic of the underlying asset is driven by a Lévy process. The performance of the proposed algorithm is evaluated by some high-dimensional PDEs on regular and irregular computational domains. Indeed, the numerical results obtained with European put options under Merton’s model show the trustability of the suggested optimal shape parameter.

Numerical simulation of 3-D fractional-order convection-diffusion PDE by a local meshless method

In this article, we present an efficient local meshless method for the numerical treatment of 3-D convection-diffusion PDE. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. This technique approximates the solution on set of uniform and scattered nodes. The space derivatives of the models are discretized by the proposed meshless procedure though the time fractional part is discretized by Liouville-Caputo fractional derivative. Some test problems on regular and irregular computational domains are presented to verify the validity, efficiency, and accuracy of the method.

A ghost structure finite difference method for a fractional FitzHugh-Nagumo monodomain model on moving irregular domain

In this paper, a ghost structure (GS) finite difference method is proposed to simulate the fractional FitzHugh-Nagumo (FHN) monodomain model on a moving irregular computational domain. In the GS formulation the moving irregular domain is converted into a fixed regular domain (called ghost structure), and the transmembrane potential is described in the Eulerian coordinates, while the membrane dynamics are described in the Lagrangian coordinates. The transformation between the Lagrangian variables and the Eulerian variables is achieved by an integral transformation which involves a delta function. The GS formulation allows to compute the transmembrane potential in a fixed regular domain using the finite difference method on a Cartesian grid, which has a huge advantage for approximating domain-dependent fractional derivatives. To overcome the difficulty caused by running time-consuming loops to compute the transformation between the Eulerian and Lagrangian variables, two fast algorithms are proposed to compute the transformation. Extensive numerical tests are provided to demonstrate the effectiveness and robustness of the proposed GS finite difference method for solving the fractional FHN monodomain model. We first numerically study the transmembrane potential propagation in both healthy hearts and hearts with arrhythmia by simulating the model in the transverse of a ventricle. We then study the transmembrane potential propagation during the pumping process, which requires to simulate the model in the moving longitudinal section of a ventricle. Our numerical results show that the change of spatial derivatives can affect the propagation velocity and the width of the transmembrane potential wave, and for a heart with arrhythmia, the transmembrane potential begins to enter cyclically the region where cardiomyocytes have been excited and then stimulates cardiomyocytes to contract again.

A meshless local moving Kriging method for solving Ginzburg–Landau equation on irregular domains

In this paper, a meshless local moving Kriging method is applied for numerical solving of nonlinear Kuramoto–Tsuzuki and Ginzburg–Landau equations on regular and irregular computational domains. These models are considered with different types of boundary conditions such as Dirichlet, Neumann and periodic boundary conditions. At first, a finite difference formula is employed for obtaining a time-discrete scheme. Then, the spatial derivatives are approximated using the moving Kriging technique. The moving Kriging method is a truly meshless method in which the unknown function can be approximated locally, and this leads to the sparsity of the coefficient matrix. Also, this method can be easily applied on irregular computational domains. All these capabilities make it an efficient method for solving the nonlinear partial differential equations especially Kuramoto–Tsuzuki and Ginzburg–Landau equations. For numerical experiments, various cases of these models are solved on regular and irregular computational domains and the efficiency of the proposed method is shown. Finally, it is concluded that the local moving Kriging method can be considered as an attractive alternative to the existing mesh-based methods in solving nonlinear Kuramoto–Tsuzuki and Ginzburg–Landau equations.

An adaptive sparse kernel technique in greedy algorithm framework to simulate an anomalous solute transport model

In the current work, an efficient and powerful computational technique is implemented to simulate an anomalous mobile-immobile solute transport process. The process is mathematically modelled as a time-fractional mobile-immobile diffusion equation in the sense of Riemann-Liouville derivative. Firstly, an implicit time integration procedure is used to semi-discretize the model in the time direction. The unconditional stability of the proposed time discretization scheme has been proven. Then an adaptive sparse meshless method has been formulated and implemented to fully discretize the model. In this approach, a kernel-based collocation method is equipped with a greedy sparse approximation procedure to discretize the governing problem on a convenient neighborhood of each data point with acceptable accuracy. Therefore, it leads to a sparse and well-conditioned algebraic system. Some test problems on regular and irregular computational domains are presented to verify the validity, efficiency, and accuracy of the method.

Numerical simulation of simulate an anomalous solute transport model via local meshless method

In this article, an efficient local meshless technique is implemented for the numerical solution of an anomalous mobile-immobile solute transport process. The process is mathematically modeled as a time fractional mobile-immobile diffusion equation in sense of Caputo derivative. An implicit time integration procedure is used to semi-discretize the model in the time direction whereas the space derivatives of the model is discretized by the proposed meshless technique based on inverse multquadric radial basis function. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. This technique approximate the solution on set of uniform and scattered nodes and it leads to a sparse and well-conditioned coefficient matrices. Numerical examinations on some test problems are performed to exhibit successful applications and accuracy of the local meshless technique on regular and irregular computational domains.

Numerical analysis of the fractional evolution model for heat flow in materials with memory

This paper develops the solution of the two-dimensional time fractional evolution model using finite difference scheme derived from radial basis function (RBF-FD) method. In this discretization process, a finite difference formula is implemented to discrete the temporal variable, while the local RBF-FD formulation is utilized to approximate the spatial variable. The pattern of data distribution in the local support domain is assumed as having a fixed number of nodes. The local RBF-FD is based on the local support domain that leads to a sparsity system and also avoids the ill-conditioning problem caused by global collocation method. The stability and convergence of time-discrete approach in H1-norm are discussed by means of the energy method. Numerical results illustrate the proposed method and demonstrate that it provides accurate solutions on regular and irregular computational domains.

Simulation of action potential propagation based on the ghost structure method

In this paper, a ghost structure (GS) method is proposed to simulate the monodomain model in irregular computational domains using finite difference without regenerating body-fitted grids. In order to verify the validity of the GS method, it is first used to solve the Fitzhugh-Nagumo monodomain model in rectangular and circular regions at different states (the stationary and moving states). Then, the GS method is used to simulate the propagation of the action potential (AP) in transverse and longitudinal sections of a healthy human heart, and with left bundle branch block (LBBB). Finally, we analyze the AP and calcium concentration under healthy and LBBB conditions. Our numerical results show that the GS method can accurately simulate AP propagation with different computational domains either stationary or moving, and we also find that LBBB will cause the left ventricle to contract later than the right ventricle, which in turn affects synchronized contraction of the two ventricles.

A meshless multiple-scale polynomial method for numerical solution of 3D convection–diffusion problems with variable coefficients

In this paper numerical solution of 3D convection–diffusion problems both with high Reynolds (Re) numbers and variable coefficients are investigated via a meshless method based on polynomial basis. It is well known that using polynomial basis directly for solving partial differential equations may be unsafe due to ill-conditioned resultant coefficients matrix that formed after discretization process. Therefore to get rid of highly ill-conditioned coefficient matrix we took advantage of multiple-scale method which is essentially based on the idea of equating norm of each column of resultant coefficients matrix. Through this approach we can reduce the condition number greatly. The proposed method is a truly meshless method since there is no need for meshing or any node connectivity and implementation of the method is simple and straightforward. The performance of the proposed method is measured by four test problems in both regular and irregular computational domains. Numerical results corroborate efficiency of meshless multiple-scale polynomial method for 3D convection–diffusion problems as well show its stability in case of large noise effect.

Hybrid absorbing boundary condition for piecewise smooth curved boundary in 2D acoustic finite difference modelling

Flexible computational domains and their corresponding irregular absorbing boundary conditions (ABCs) have previously been shown to improve the efficiency of finite difference (FD) modelling. However, these proposed ABCs for irregular boundaries are based on irregular grid methods. Although they can improve geometric flexibility of FD modelling, irregular grid methods are still complex to implement due to computationally expensive meshing process. To avoid complex mesh generation, FDs in mesh-free discretisation have been developed for nontrivial geometric settings, where scattered nodes can be placed suitably with respect to irregular boundaries and arbitrarily shaped anomalies without coordinate mapping or mesh-element forming. Radial-basis-function-generated FD (RBF-FD) has been proven successful in modelling seismic wave propagation based on mesh-free discretisation. Using RBF-FD, we develop a hybrid ABC for piecewise smooth curved boundary in 2D acoustic wave modelling based on straightforward expanding strategies for sampled boundary and corner nodes. The whole irregular computational domain is combined by an objective zone and a transition zone separated by piecewise smooth curved boundary. Nodal distribution in the objective zone is adaptive to the model structure through special alignment and varying density of nodes, while the transition zone are designed for solving one way wave equation more easily and accurately in the presence of piecewise smooth curved boundary. Modelling examples for homogenous and heterogeneous models with differently shaped computational domains demonstrate the effectiveness of our proposed method.

An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional Bloch–Torrey equations

The main aim of the current paper is to propose an efficient numerical technique for solving two-dimensional space-multi-time fractional Bloch–Torrey equations. The current research work is a generalization of [6]. The temporal direction is based on the Caputo fractional derivative with multi-order fractional derivative and the spatial directions are based on the Riemann–Liouville fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order O(τ2−α). Also, the space variable is discretized using the finite element method. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, four test problems have been illustrated to verify the efficiency and simplicity of the proposed technique on irregular computational domains.

Numerical simulations of viscoelastic flow in complex geometries using a multi-mode Giesekus model

We present a stable and convergent numerical method to simulate unsteady incompressible viscoelastic flow in two dimensional complex geometry. The incompressible viscous momentum equation, coupled with the multiple mode Giesekus constitutive equation for viscoelastic stress, are used to model viscoelastic fluids. We cast the hyperbolic part of the equations as a conservative finite difference method and leave the elliptic part as a source term computed by implicit methods. A projection method is used to enforce the velocity incompressibility constraint and to update pressure. Irregular computational domains are discretized with a Cartesian grid embedded boundary method. We demonstrate our numerical method by computing flow passing a sphere, flow in a well-rounded contraction channel, and in an abrupt contraction channel, with component Weissenberg numbers exceeding 300, and with a range of relaxation times spanning four decades.

Numerical Analysis of General Groove Geometry for Dry Gas Seals

By Changing the key points on the spiral curve, general groove geometry was determined. Considering the simplicity of modeling and analysis, cubic spline function was used to express the general groove profile. By using the boundary fitted coordinate system transformation, irregular computational domain was transferred to regular region; Based on flow conservation principle, finite volume method was applied to discrete compressible Reynolds equation; By the application of Newton-Raphson iteration method for solving algebraic equation, numerical model of general groove dry gas seals was established. When compared sample results with shallow groove theory, the capacity and stiffness of numerical results match well with theoretical ones, verifying the accuracy of novel numerical model. Through analysis of three typical groove seals, spiral groove seal has strongest carrying capacity. Pressure distribution of three groove seals subjects to the law of hydrodynamic pressure effect. And the numerical model established in this paper will offer a general calculate platform for optimization of groove geometry in the future.

Steady state conduction through 2D irregular bodies by smoothed particle hydrodynamics

This paper uses a Lagrangian formulation namely smoothed particle hydrodynamics (SPH) technique for the prediction of conduction heat transfer in irregular geometries. Suitable schemes for the placement of particles inside irregular computational domain have been described. Organization and locations for virtual particles are also discussed for both Dirichlet and Neumann boundary conditions. Three different problems having typical geometries that pose numerical complications in grid based technique, are tackled by SPH efficiently. Conduction shape factor (CSF) for all the problems calculated through SPH shows a good agreement with published results or analytical solutions. Further, the temperature profiles computed by SPH depict close matches with those obtained through numerical software Fluent. The present exercise opens up the possibility of using SPH in varied geometries using different schemes of particle positioning and their judicial combinations.