Complex Computational Domains Research Articles

Zonal free element method for free and forced vibration analysis of two- and three-dimensional structures

This paper presents a new numerical method called the zonal free element method (ZFREM) for the free and forced vibration analysis of elastodynamic problems. In this approach, a complex computational domain is divided into some simple zones and generates a series of regularly arranged nodes in each zone, which can improve the accuracy during the analysis of complex models. The distinguishing feature of the ZFREM is that an independent isoparametric element is formed by only one freely chosen surrounding node at each configuration node. In this method, the mass term exists only in the internal nodes, which can accelerate the assembly of the final system equations. Building upon this foundation, the present study developed the Krylov reduced dimensional iterative method, which approximates the solution of the equation system by constructing a lower-dimensional subspace. This approach avoids the complex equation transformations involved in traditional algorithms for solving eigenvalue problems, thereby further enhancing computational efficiency. Moreover, to tackle damped vibration problems encountered in engineering applications, the proposed method is further extended to solve the non-linear forced vibration problems. The accuracy and effectiveness of the method are verified by numerical examples of free and forced vibration problems.

Comparison of Hexcore and Poly-Hexcore computational meshes in the aspect of air flow modeling based on the actual geometry of mining excavations

Discrete models are used in industry for many applications. In one of the most frequently used Finite Element Method (FEM) for Computational Fluid Dynamics (CFD) calculations, discrete models may be two-dimensional or three-dimensional. 2D models are used as a simplification to achieve satisfied results in the shortest computational time. 3D models, on the other hand, are used for more complex calculations. These models constitute a representation of real-world objects that have been appropriately simplified to make the calculations accurate and correct. The calculation time of a 3D model is significantly longer compared to a 2D model. For this reason, to reduce the calculation time, different types of simplifications and various types of discrete model meshes are used. In this paper, the authors made a comparison of two computational meshing technologies: Hexcore and Poly-Hexcore in the aspect of airflow modeling in mining excavations using CFD. The geometry considered in this case came from real-world excavations captured by laser scanning in the Gertruda Slant, Zloty Stok. Point cloud data was processed through feature extraction, which was subsequently utilized to create structured models of mining excavations. The results of the simulations show that taking into account such a diverse and complicated geometry and its significant lengths, reaching tens of kilometers, better results are obtained with the use of Poly-Hexcore mesh. This type of mesh allows simulations to be performed with similar accuracy in a shorter computation time. Utilizing a more modern type of mesh makes work more dynamic, which is of particular importance when conducting numerical simulations of air distribution in large and complex computational domains.

An accurate RBF–based meshless technique for the inverse multi-term time-fractional integro-differential equation

Using the Grünwald difference operator one reduces the inverse boundary value problem of the multi-term time-fractional integro-differential equation (TFIDE) in two dimensions to a system L[u]=F. The backward substitution method (BSM) which is based on the trigonometric basis function (TBF) is developed to a new version and examined for the system. The solution of ill-conditioned TFIDE is expressed in terms of the TBFs on the corrupted boundary data with a small parameter in the noisy data. Finally, the TFIDE model in the complex computational domain is assessed through the methodology with reasonable selective parameters, in addition, high-accuracy solutions and acceptable estimation for the errors are illustrated.

A fast inversion approach for the identification of highly transient surface heat flux based on the generative adversarial network

The efficient identification of surface heat flux during highly transient pool boiling process is an essential task for engineering production and manufacture, which can be mathematically defined as nonlinear inverse heat transfer problems (IHTP) of Neumann boundary conditions. Conventional regularization-based inversion approaches for these types of problems are computationally expensive since packs of nonlinear partial differential equations (PDE) constrained optimization problems in complex computational domains need to be solved. In this paper, an operator-learning method is proposed for the efficient solution of complex three-dimensional (3D) transient nonlinear IHTP. FluxNet, a generated adversarial network composed of a discrete wavelet transform (DWT) layer, a “generator” and a “discriminator”, is constructed to significantly improve the computational efficiency, reaching sub-second for the first time. The DWT layer is capable of extracting detailed information of the temperature distribution while the “generator” approximates the inverse operator of the governing PDE system with its convolution kernels. A minimax optimization strategy is adopted through adversarial training between the “generator” and the “discriminator” to encourage deeper understanding of data distribution. Compared with other types of deep learning methods, the proposed method shows its superiority when dealing with highly transient experimental data and micro-nano porous structured geometry. Based on the developed sub-second light-weight numerical solver with operator learning, the development of advanced real-time soft sensor techniques for a class of 3D transient IHTP in practical engineering applications is possible.

Application of generalized finite difference method for elastoplastic torsion analysis of prismatic bars

In this paper, the generalized finite difference method (GFDM) is employed for the elastic-plastic torsion analysis of prismatic bars, and the generalized finite difference numerical scheme of the elastic-plastic torsion governing equation is derived. This approach divides the computational domain into a series of overlapping sub-domains and constructs a system of linear equations based on Taylor series expansion and moving least squares. With the help of Picard's iterative method, the elastic solution is utilized as the initial guess to reach the elastic-plastic solution. The plastic nonhomogeneous term is then evaluated via radial basis function (RBF) interpolation. Since the generalized finite difference method does not require intra-domain integration and allows random distribution of nodes, it could simply deal with complex computational domain problems. In numerical experiments, the elastic-plastic torsion examples of square-section bars and linear guide rail reveal that the numerically developed generalized finite difference model is a reliable and effective analysis solution for elastic-plastic torsion problems of prismatic bars.

Meshfree methods for nonlinear equilibrium radiation diffusion equation

The radiation diffusion problem is usually applied to astrophysics and inertial confinement fusion. It seems that the traditional numerical methods such as finite difference method and finite element method is not efficiency to solve the problem because of its nonlinearity and complex computational domain. In this paper, some fast meshfree methods based on radial basis function are proposed to solve the equilibrium radiation diffusion equation. Because the modeling problem has two nonlinear terms, so we use the successive permutation iteration and linearization methods to approximate the nonlinear terms. Firstly, diffusion terms of the one-dimensional (1D) and two-dimensional (2D) equilibrium radiation diffusion equations are linearized by constructing a successive permutation iteration method for fully-implicit discretization. Then, three kinds of different linearization methods (factorization iteration, Picard–Newton iteration and Richtmyer iteration) are proposed and used to linearization of other nonlinear terms. Finally, the linearized equation is solved numerically by Kansa’s method with compactly supported radial basis function. The reliability, accuracy and effectiveness of new methods are verified by 1D and 2D numerical experiments. The advantages of presented method are apparent. They require neither domain nor boundary discretization, and no domain integration is required. New methods are easy for implementation and coding. They not only avoid the complexity of generation mesh, but also solve the equilibrium radiation diffusion equation with high accuracy.

A regularized diffuse domain-lattice Boltzmann model for heat transfer in complex geometries with temperature Dirichlet boundary condition

In this paper, a regularized diffuse domain-lattice Boltzmann model for heat transfer with the Dirichlet boundary condition in complex geometries is proposed. In this model the complex computational domain is embedded into a larger and regular domain. The level set function is used to capture the complex interface. A diffuse domain formulation for the temperature field is established on the larger domain. To remove the ill-conditioned property of the original diffuse domain equation, we develop a new regularization technique to tackle this issue. We also give a multiple-relaxation-time lattice Boltzmann model to solve the regularized diffuse domain equation. The convergence rate of present model is investigated by the steady state and time-decay heat conduction problems. Then the natural convection problems in a square enclosure with a heated circular cylinder and in a horizontal concentric annulus are simulated. Both the uniform and non-uniform wall temperature cases are considered. Moreover, an alternative method to calculate the average Nusselt number on the complex boundary is employed. The numerical results show a good agreement with the previous solutions in previous literatures. The effect of non-uniform wall temperature on the heat transfer rate is also investigated. At last, the forced convection across the staggered tube banks is also simulated to verify the potential capability of the present model in solving practical heat transfer problem with the complex geometries.

Deep Learning for Computational Hemodynamics: A Brief Review of Recent Advances

Computational fluid dynamics (CFD) modeling of blood flow plays an important role in better understanding various medical conditions, designing more effective drug delivery systems, and developing novel diagnostic methods and treatments. However, despite significant advances in computational technology and resources, the expensive computational cost of these simulations still hinders their transformation from a research interest to a clinical tool. This bottleneck is even more severe for image-based, patient-specific CFD simulations with realistic boundary conditions and complex computational domains, which make such simulations excessively expensive. To address this issue, deep learning approaches have been recently explored to accelerate computational hemodynamics simulations. In this study, we review recent efforts to integrate deep learning with CFD and discuss the applications of this approach in solving hemodynamics problems, such as blood flow behavior in aorta and cerebral arteries. We also discuss potential future directions in the field. In this review, we suggest that incorporating physiologic understandings and underlying fluid mechanics laws in deep learning models will soon lead to a paradigm shift in the development novel non-invasive computational medical decisions.

IGA-suitable planar parameterization with patch structure simplification of closed-form polysquare

A primary challenge for isogeometric analysis (IGA)-suitable parameterization is to efficiently decompose a complex computational domain into a small number of high-quality IGA-suitable patches without internal singularities. Aiming at domains with arbitrary topology, we propose integrating frame field and polysquare structure, resulting in a structure associated with a frame field that is locally integrable everywhere, i.e. a closed differential form or closed-form. Such a structure is more general than a common polysquare but has no internal singularity as well, which makes it suitable for IGA because it usually has fewer undesired boundary corners and low distortion. Since IGA prefers a small number of patches in the parameterization structure, we further propose a patch simplification method with distortion control. A key challenge in simplification is to prevent the resulting patches from degenerating. Previous methods rely on a mesh of the domain to characterize the non-degenerate condition. In sharp contrast, our approach simply uses the boundary of the domain, which obviously improves efficiency because the number of variables is much lower. The patch degeneration issue in such a boundary-only strategy is addressed by solving a constrained optimization problem with a set of automatically constructed inequality constraints. We prove that such constraints are sufficient to make the common polysquare structure homeomorphic to the input shape. We also demonstrate that it prevents the patches from degenerating for the structure of closed-form polysquares, and it can be extendedly applied to 3D closed-form polycubes. Numerical experiments using bi-quadratic B-spline surfaces with C1-constraints are also provided to justify the accuracy of applying the resulting planar parameterization for IGA application.

Integrated radial basis functions to simulate modified anomalous sub‐diffusion equation

AbstractThis research work presents a newly truly meshless approach based upon the integrated radial basis functions (IRBFs) technique to study the fractional modified anomalous sub‐diffusion equation. First, the temporal direction is discretized by a finite difference method with second‐order accuracy. The stability analysis and convergence of the proposed time‐discrete formulation are investigated, theoretically. Then, the spatial direction is approximated by the IRBFs methodology. In this approach, the largest‐order derivative is approximated by a linear combination of RBFs and then the lower‐order derivative and also the unknown function are constructed by repeated integration. According to this procedure, the existing derivatives in the main fractional PDE are approximated smoothly. Furthermore, to show the efficiency of the developed numerical procedure, we employed some non‐rectangular computational domains to obtain the numerical results. On the other hand, the mentioned model is solved in one‐ two‐, and three dimensional cases. The numerical results confirm the ability and efficiency of the new numerical method for solving time fractional PDEs on complex computational domains.

A simple and explicit numerical method for the phase-field model for diblock copolymer melts

In this paper, we present a simple and explicit finite difference method for the phase-field model for diblock copolymer melts. A diblock copolymer is a polymer consisting of two types of different monomers bonded covalently to each other to form a single copolymer chain. When the temperature is below the critical temperature, the copolymer melt exhibits microphase separation. The mathematical model is derived from a total free energy functional which contains kinetic, gradient, double well, and long-range nonlocal potentials. The Saul’yev-type scheme based on a linearly stabilized convex splitting method is used for the discretizations. The proposed method is simple and computationally efficient because the scheme is explicit and it does not require any iterative procedures. The proposed scheme not only overcomes the severe time step restriction for the explicit scheme but also works well for the simulations of lamellar and hex-cylinder structures which are characteristic morphologies for diblock copolymer melts after phase separation. Furthermore, the proposed method can be easily applied to the simulations in complex computational domains. We present various numerical tests to demonstrate the performance of the proposed scheme.

A NEURAL SOLVER FOR VARIATIONAL PROBLEMS ON CAD GEOMETRIES WITH APPLICATION TO ELECTRIC MACHINE SIMULATION

This work presents a deep learning-based framework for the solution of partial differential equations on complex computational domains described with computer-aided design tools. To account for the underlying distribution of the training data caused by spline-based projections from the reference to the physical domain, a variational neural solver equipped with an importance sampling scheme is developed, such that the loss function based on the discretized energy functional obtained after the weak formulation is modified according to the sample distribution. To tackle multi-patch domains possibly leading to solution discontinuities, the variational neural solver is additionally combined with a domain decomposition approach based on the Discontinuous Galerkin formulation. The proposed neural solver is verified on a toy problem and then applied to a real-world engineering test case, namely that of electric machine simulation. The numerical results show clearly that the neural solver produces physics-conforming solutions of significantly improved accuracy.

Solve Laplace-Beltrami Equation on Geometrically Continuous Curves in Isogeometric Analysis

<p indent=0mm>Research on isogeomatric analysis on a complex computational domain is one of the key problems. Normally a complex computational domain is composed by several simple patches with geometric continuity. Thus, it is necessary to discuss the relationship between the convergence of isogeometric analysis and geometric continuity. Solving Laplace-Beltrami equation on a geometrically continuous curve by analysis isogeometric analysis error theory is discussed. Based on this theoretic result, a method for choosing a spline space to obtain an optimal convergence rate in isogeometric solving is introduced and the numerical experiments are illustrated. Moreover, we validate the spline space, chosen by its approximate property, reaches the optimal convergence rate numerically. The results provide a theoretical basis for isogeometric analysis on complex computational domains.

Localized singular boundary method for the simulation of large-scale problems of elliptic operators in complex geometries

In this paper, the novel localized singular boundary method (LSBM) in conjunction with the Chebyshev collocation scheme (CCS) is developed to simulate large-scale inhomogeneous problems governed by elliptic operators. By means of the Gauss–Lobatto collocation points, the CCS is employed to obtain the particular solution. With the obtained particular solution, the inhomogeneous elliptic boundary value problem can be transformed into a classical homogeneous problem which can be solved by the LSBM. In the LSBM, the whole computational domain with distributed nodes is divided into a series of overlapping circle-shaped sub-domains. For each sub-domain, the moving least square method and the conventional singular boundary method (SBM) are utilized. Considering that the matrix generated by the CCS-LSBM is sparse, this scheme is granted the tremendous potential to address large-scale problems. Numerical experiments are presented to verify the effectiveness and accuracy of the proposed scheme for problems with mixed boundary conditions, sophisticated analytical solutions, and complex computational domains, respectively.

A novel combined space-time algorithm for transient heat conduction problems with heat sources in complex geometry

A space-time algorithm, integrated by the localized space-time method of fundamental solutions (LSTMFS) and space-time Chebyshev collocation scheme (STCCS), is developed to resolve heat conduction problems with heat sources. Based on the Chebyshev collocation scheme, a STCCS involving Gauss-Lobatto space-time nodes is constructed to obtain the particular solution of the inhomogeneous problem. The particular solution is then used to convert the inhomogeneous transient heat conduction problem to a classical homogeneous one which can be directly solved via the LSTMFS. Considering that no temporal-difference is employed and the generated matrix is sparse, the STCCS-LSTMFS has promising potential to directly and quickly address inhomogeneous transient heat conduction problems. Numerical experiments are presented to show the accuracy and efficiency of the proposed methodology for the transient heat conduction problems with sophisticated/unknown analytical solution, mixed boundary conditions, and complex computational domain.

Local radial basis function collocation method for stokes equations with interface conditions

In the present work, local radial basis function (RBF) collocation method is used to solve the Stokes problem numerically. Benefits of the local meshless method is its flexibility of handling complex computational domain and interface. Local meshless method produces sparse coefficient matrix, though its structure is depending on the placement of nodes, and type of boundary conditions. The Stokes equations are collocated in a rectangular region with different interface conditions and direct solvers are used instead of iterative algorithm. Due to the unknown boundary conditions for the pressure term in the Stokes equations, value of the pressure is obtained by integrating the Stokes equations and eliminating the constant of integration. In the numerical implementation, circular, elliptic and Deltoid like interfaces are taken into consideration to test performance of the proposed approach for complex problems. Meshless solution of the Stokes equations for the first two problems are compared with the published results in the literature. The last problem is different form the first two problems in the sense that the continuity equation is also discontinuous and the velocity has nonzero value in both the inner and outer sub-domains.

Sneezing and asymptomatic virus transmission.

The novel coronavirus disease (COVID-19) spread pattern continues to show that geographical barriers alone cannot contain a virus. Asymptomatic carriers play a critical role in the nature of this virus quickly escalating into a global pandemic. Asymptomatic carriers may transmit the virus unintentionally through sporadic sneezing. A novel Computational Fluid Dynamics (CFD) approach has been proposed with a realistic modeling of a human sneeze achieved by the combination of state-of-the-art experimental and numerical methods. This modeling approach may be suitable for future engineering analyses aimed at reshaping public spaces and common areas, with the main objective to accurately predict the spread of aerosol and droplets that may contain pathogens. This study shows that the biomechanics of a human sneeze, including complex muscle contractions and relaxations, can be accurately modeled by the angular head motion and the dynamic pressure response during sneezing. These have been considered as the human factors and were implemented in the CFD simulation by imposing a momentum source term to the coupled Eulerian–Lagrangian momentum equations. The momentum source was modeled by the measured dynamic pressure response in conjunction with the angular head motion. This approach eliminated the need to create an ad hoc set of inlet boundary conditions. With this proposed technique, it is easier to add multiple fixed and/or moving sources of sneezes in complex computational domains. Additionally, extensive sensitivity analyses based on different environmental conditions were performed, and their impact was described in terms of potential virus spread.

An accurate triangular spectral element method-based numerical simulation for acoustic problems in complex geometries

An accurate triangular spectral element method (TSEM) is developed to simulate acoustic problems in complex computational domains. With Fekete points and Koornwinder-Dubiner polynomials introduced, triangular elements are used in the present method to substitute quadrilateral elements in traditional spectral element method (SEM). The efficiency of discretizing complex geometry is enhanced while high accuracy of SEM is remained. The weak form of the second-order governing equations derived from the linearized Euler equations (LEEs) are solved, and perfectly matched layer (PML) boundary condition is implemented. Three benchmark problems with analytical solutions are employed to testify the exponential convergence rate, convenient implementation of solid wall boundary condition and capable discretization in complex geometries of the present method respectively. An application on Helmholtz resonator (HR) is presented as well to demonstrate the possibility of using the present method in practical engineering. The numerical resonance frequency of HR reaches an excellent agreement with the theoretical result.

An improved method for the calculation of unsaturated\u2013saturated water flow by coupling the FEM and FDM

Numerical modeling of water movement in both unsaturated soils and saturated groundwater aquifers is important for water resource management simulations. The development of efficient numerical algorithms for coupling unsaturated and saturated flow has been a long-lasting challenge in hydrologic modeling, especially for regional-scale modeling. In this study, a new method coupling the Finite Element Method (FEM) and Finite Difference Method (FDM), FE-FDM, is developed to solve Richards equation for simulating unsaturated–saturated water flow. The FEM is adopted to discretize the governing equation in the horizontal direction, while the FDM is used in the vertical direction. This method combines the advantages of FEM in domain discretization, especially for complex computational domain, and the advantages of FDM in modeling simplicity and efficiency. The validity of the new method is demonstrated with three test cases that show that the FE-FDM solutions are accurate and are applicable for regional scale problems. In the test cases, the FE-FDM solutions are compared with experimental data and numerical results calculated with common software packages including FEFLOW and COMSOL. This study verified that the FE-FDM is applicable for simulating water flow in the unsaturated–saturated zone.

Hydrodynamic forces and heat transfer of nanofluid forced convection flow around a rotating cylinder using finite element method: The impact of nanoparticles

A computational approach is utilized to study the nanofluid forced convection flow around a rotating circular cylinder in a horizontal channel. Different types of nanoparticles in the base fluid are considered and investigated such as Silica (SiO2), Copper (Cu) and Alumina (Al2O3). Average Nusselt number (Nuavg) are compared for various conditions of the cylinder, namely adiabatic and isothermal cylinders respectively for two locations of cylinder. A dimensionless system of partial differential equations over a complex computational domain is discretized by applying the higher order stable finite element method. The discretized nonlinear system of algebraic equations is linearized using the Newton method. A geometric multigrid method is implemented for the computation of linearized subproblems in each nonlinear sweep. Impact of the pertinent parameters is investigated such as the Reynolds number (10 ≤ Re ≤ 200), angular velocity (−75 ≤ Ω ≤ 75) and the nanoparticles volume fraction (0.0 ≤ ϕ ≤ 0.04). Numerical results are demonstrated in the form of isotherms, streamlines and suitable graphs for various quantities of interest. It is inferred that the clockwise rotation of the cylinder makes the fluid move over the cylinder while the fluid moves below the cylinder in the anticlockwise case. Moreover, an increase in the nanoparticles volume fraction enhances the average Nusselt number and decreases both of the lift and drag coefficients.