Unstructured Triangular Elements Research Articles

Flooding study of the Loira River (Galicia, Spain): Importance of pre-evaluation in land management

Floods pose recurrent threats to numerous countries, carrying substantial social, economic and at times, catastrophic implications, especially in extreme scenarios. In addition, coastal areas are strongly threatened by extreme weather events and climate change impacts. Given these pressing challenges, it becomes imperative to conduct thorough assessments of future flood impacts, particularly in areas of significant environmental and socio-economic importance. This study addresses this need by focusing on the potential consequences of flooding in the Loira River, situated in the municipality of Marín, Pontevedra, Northwestern Spain. For this purpose, the free software IBER has been used, these is a free software that solves the 2D depth-averaged shallow water equations (SWEs) by using a finite volume scheme, with the domain being discretized with both structured and unstructured triangular or quadrilateral elements. The hydrological model has been established from the maximum flow for different return periods (T10, T100 and T500). Field observations, including channel widths, bridge configurations, and various parameters, further inform our analysis. Our findings reveal critical insights, including a maximum height differential of 0.48 m, particularly significant at the upstream bridge, exacerbating flooding risks in adjacent plains. Moreover, velocities along riverbanks reach hazardous levels, notably during the 500-year return period, necessitating urgent protective measures for local residents. The hydrological model also indicates that small-scale forestation has no significant effect on flood prevention. This research underscores the utility of advanced hydrological modelling in informing risk management strategies and underscores the necessity of pre-emptive hydrological analyses to mitigate future flood risks effectively.

Wavefield simulation with the discontinuous Galerkin method for a poroelastic wave equation in triple-porosity media

In this paper, we derive a new poroelastic wave equation in triple-porosity media and develop a weighted Runge-Kutta (RK) discontinuous Galerkin method (DGM) for solving it. Based on Biot’s theory and Lagrangian formulas, we obtain 3D Biot’s equations in a heterogeneous anisotropic triple-porosity medium. We also summarize poroelastic wave equations in single- and double-porosity media. The traditional two-phase theory is a special case of the porous theory. Compared with single- or dual-porosity wave equations, our triple-porosity wave equation generates more accurate wavefield information. The isotropic and anisotropic cases are considered. Subsequently, we formulate the new poroelastic equation into a first-order hyperbolic conservation system, which is suitable to be solved by DGM. An optimized local Lax-Friedrichs flux and an implicit-weighted RK time discretization scheme are used for this computation. We use two types of mesh elements — quadrilateral and unstructured triangular elements. We find that there are two kinds of slow P waves, P1 and P2 waves, in double-porosity media, whereas three kinds of slow P waves, P1, P2, and P3 waves, exist in three-porosity media. We also study the analytical and numerical solutions of propagation velocities for different waves in isotropic media without dissipation using the Jacobian matrix of DGM and provide a comparison of field variables about three types of wave equations. Finally, we conduct a series of examples to quantitatively investigate the propagation properties of seismic waves in isotropic and anisotropic multiporosity media computed by DGM. The slow P wave in multiporosity media with dissipation decays rapidly, which also will lead to phase distortion. Numerical results verify the correctness and applicability of our proposed new equation and indicate that the weighted RK DGM is a stable and accurate algorithm to simulate wave propagation in poroelastic media.

Numerical solution of magnetohydrodynamics effects on a generalised power law fluid model of blood flow through a bifurcated artery with an overlapping shaped stenosis.

This paper presents a numerical analysis of blood flow in a diseased vessel within the presence of an external magnetic field. The blood flow was considered to be incompressible and fully developed, in that the non-Newtonian nature of the fluid was characterised as a generalised power law model for shear-thinning, Newtonian, and shear-thickening fluids. The impact of a transverse directed external magnetic field on blood flow through a stenosed bifurcated artery was investigated. The arterial geometry was considered as a bifurcated channel with overlapping shaped stenosis. The problem was treated mathematically using the Galerkin Least-Squares (GLS) method. The implementation of this numerical method managed to overcome the numerical instability faced by the classical Galerkin technique when adopted to a highly viscous flow. The benefit of GLS in circumventing the Ladyzhenskaya-Babuška-Brezzi (LBB) condition was utilized by evaluating both the velocity and pressure components at corner nodes of a unstructured triangular element. The non-linearity that emerged from the convective terms was then treated using the Newton-Raphson method, while the numerical integrals were computed using a Gaussian quadrature rule with six quadrature points. The findings obtained from this study were then compared with available results from the literature as well as Comsol multiphysics software to verify the accuracy and validity of the numerical algorithms. It was found that the application of magnetic field was able to overcome flow reversal by 39% for a shear-thinning fluid, 26% for a Newtonian fluid, and 27% for a shear-thickening fluid. The negative pressure and steep wall shear stress which occurs at the extremities of an overlapping stenosis throat were diminished by rise in magnetic intensity. This prevented thrombosis occurrence and produced a uniform calm flow.

A ghost-node immersed smoothed point interpolation method (ghost-node-ISPIM) for fluid-structure interaction problems

Immersed smoothed point interpolation method (IS-PIM) was successfully proposed for solving fluid-structure interaction (FSI) problems with large solid deformations. However, the defects of IS-PIM include inaccurate solid boundary, spurious pressure oscillation and the “fresh node” phenomenon, which will cause different degrees of numerical errors. In order to solve these problems, the ghost-node immersed smoothed point interpolation method (Ghost-node-ISPIM) has been presented in this paper. In this method, the ghost-node technique is firstly proposed and achieved based on unstructured triangular elements, combined with mass source/sink algorithm and the sharp-interface method to fix the inherent defects existed in original IS-PIM. The effectiveness and superiority of the Ghost-node-ISPIM are demonstrated by a series of typical FSI numerical examples.

MATLAB PDE toolbox for 2-D and 3-D neutron diffusion simulations

This research extends MATLAB PDE Toolbox to model nuclear reactors by solving the 2-D and 3-D multigroup neutron diffusion equations using the finite element method. The developed scripts can simulate reactors with rectangular and hexagonal lattices and have the potential to handle complex core configurations, benefitting from the constructive solid geometry. The meshing subroutine decomposes the domain into unstructured triangular or tetrahedral elements that are either linear or quadratic. With many versatile MATLAB functions, data analysis and visualization can be accomplished within the same software environment. The proposed toolbox provides a graphical user interface for novices and simple models and a programmatic workflow for experienced users and complex problems. Multiple benchmark tests show that the developed framework produces accurate results with acceptable computation speed. The developed MATLAB reactor simulation engine is license friendly and readily accessible by researchers and students, making it an ideal research and educational tool.

Optimizing the velocity of ring shape parameter for designing the nozzles using CFD

This study aims to optimize the velocity of ring shape parameter for designing the nozzles using computational fluid dynamics (CFD) and investigated the flow in nozzles using ANSYS, Inc. simulation software. The model geometries were defined using ANSYS FLUENT-Design Modeler platform. All nozzles were designed on unstructured triangular elements comprising of 1200000 mesh nodes. The differential governing equations were applied in ANSYS FLUENT based on a finite volume method. The distance and dimensions of ring location significantly influence the velocity of water during flow where the maximum velocity at double rings reduces the surface area at distance of 7mm and 15mm and 2x2 mm dimensions. Considering 8, 10, and 12 bar liner proportions, there was an increase in the velocity at maximum points in ring shapes.

A Limit Equilibrium Model of Tabular Mine Pillar Failure

An improved understanding of pillar strength and pillar failure mechanisms is required to optimize tabular mine layout designs. The paper describes the application of a limit equilibrium model for pillar failure analysis. It is shown that the model is capable of reproducing a hardening or softening response for uniformly compressed strip or square pillars. The stress–strain behavior depends on three non-dimensional parameters Q, M and β. Q represents the ratio of the limit failed uniaxial strength to the intact material strength, M is the ratio of the failed limit strength envelope slope to the intact strength envelope slope and β is proportional to the pillar width to height ratio. The model is implemented in a displacement discontinuity solution scheme using unstructured triangular elements to allow irregular plan-view pillar shapes and mining step increments to be represented. The seam-parallel confining stress distribution in the fracture zone is determined using a fast marching solution algorithm. A case study of an experimental pillar extraction site in a platinum mine is presented to illustrate the capability of the model to simulate the evolution of pillar failure as pillar extraction proceeds. Good qualitative agreement to observed failure trends are obtained but the detailed calibration of the model parameters remains a challenge. Further work is required to enhance the representation of pillar edge spalling processes.

An unstructured mesh algorithm for simulation of hydraulic fracture

The paper describes a flexible computational method that can be used to represent the complex geometric evolution of a fluid-driven fracture front using an unstructured triangular element mesh. A specific motivation for the method is to allow subsequent treatment of non-planar fracture growth problems. Elastic interactions between the crack opening displacements and the surrounding medium are calculated using displacement discontinuity boundary element influence functions. A novel feature of the approach is the use of an adjustable region of tip elements to accommodate both time-dependent crack edge movements and to ensure correct velocity-dependent asymptotic behavior for viscous and viscous-toughness flow conditions. The adjustable tip element region is represented using a special-purpose data structure that allows the iteration of the moving edge element vertices to be linked directly to the flow rate and the crack opening solution values that are determined in each time step. The fringe region is periodically replaced with a set of fixed vertex elements as the fracture surface evolves. The proposed scheme has been evaluated using available analytic and experimental results for planar fracture propagation. The method is found to be both accurate and robust for a range of choices of geometric fracture shapes and time step sizes.

A coarse-grid incremental pressure projection method for accelerating low Reynolds number incompressible flow simulations

Coarse grid projection (CGP) multigrid techniques are applicable to sets of equations that include at least one decoupled linear elliptic equation. In CGP, the linear elliptic equation is solved on a coarsened grid compared to the other equations, leading to savings in computations time and complexity. One of the most important applications of CGP is when a pressure correction scheme is used to obtain a numerical solution to the Navier-Stokes equations. In that case there is an elliptic pressure Poisson equation. Depending on the pressure correction scheme used, the CGP method and its performance in terms of acceleration rate and accuracy level vary. The CGP framework has been established for non-incremental pressure projection techniques. In this article, we apply CGP methodology for the first time to incremental pressure correction schemes. Both standard and rotational forms of the incremental algorithms are considered. The influence of velocity Dirichlet and natural homogenous boundary conditions in regular and irregular domains with structured and unstructured triangular finite element meshes is investigated. $L^2$ norms demonstrate that the level of accuracy of the velocity and the pressure fields is preserved for up to three levels of coarsening. For the test cases investigated, the speedup factors range from 1.248 to 102.715.

Proposing an optimal tree-like design of highly conductive material configuration with unequal branches for maximum cooling a heat generating piece

Due to the concerns regarding the space and cost of high thermal conductivity materials, researching for a better design of high thermal conductivity pathways embedded into a heat generating piece is important. Constructal design theory is applied and a numerical optimization procedure is carried out to find a new geometric pattern of tree-like highly conductive material pathways with unequal branches. The optimization purpose is the minimization of the peak temperature in the heat generating piece, by varying the lengths of the branches and the initial distance from the middle plane. The constraint is the fixed volume fraction occupied by the high thermal conductivity insert. Heat conduction equations are solved by a finite element method (FEM). The numerical code develops in MATLAB software and use unstructured triangular elements for meshing. Because of the multiple optimization variables existed in this work, the “Pattern Search” algorithm is used. The results indicate that the performance of the tree-like configuration with unequal branches is better than the one with equal branches for the same number of branches, the same volume fraction of the highly conductive material and the same conductive ratio. It is also shown that the new proposed configurations of highly thermal conductive pathways, significantly surpass the latest configurations presented in the literature. Depending on the volume fraction of highly conductive material and thermal conductivity ratio, the performance of the proposed configurations is observed to be 2–61% better than the performance of the best result existed in previous works.

Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows

This paper is concerned with the numerical simulation of time-dependent variably-saturated Darcian flow problems described by the Richards equation. We present the adaptive higher-order space-time discontinuous Galerkin (hp-STDG) method which optimizes accuracy and efficiency by balancing the errors that arise from the space and time discretizations and from the resulting nonlinear algebraic system. Convergence problems related to the transition between unsaturated flow and saturated flow are eliminated by regularizing the constitutive formulas. We also present an hp-anisotropic mesh adaptation technique capable of generating unstructured triangular elements with optimal sizes, shapes, and polynomial approximation degrees. Several numerical experiments are presented to demonstrate the accuracy, efficiency, and robustness of the numerical method presented here.

Matrix-Free Method for Transient Maxwell-Thermal Cosimulation in Arbitrary Unstructured Meshes

Existing electrical–thermal cosimulation methods require solving a system matrix when handling inhomogeneous materials and irregular geometries discretized into unstructured meshes. In this paper, a matrix-free method is developed for cosimulating full-wave Maxwell’s equations and the thermal diffusion equation in the time domain. The method is free of matrix solutions regardless of the shape of the element used for space discretization. A theoretical stability analysis is also developed for the coupled electrical and thermal analysis, which is nonlinear in nature. Numerical experiments on both unstructured tetrahedral and triangular prism element meshes have validated the accuracy and efficiency of the proposed method.

Finite element computational procedure for convective flow of nanofluids in an annulus

In the present study, the detailed procedures of Galerkin weighted residual technique of finite element method (FEM) for solving two-dimensional incompressible natural convective flow of nanofluids using nonhomogeneous dynamic model are discussed for the first time. The physical domain is discretized by using unstructured triangular elements. The governing partial differential equations of nanofluids are made dimensionless using the suitable transformation of variables for weak formulations. The method of weighted residuals is used for obtaining the approximate solutions. This approach typically leads to a sparse and indefinite matrix that is difficult to solve efficiently. The formation of an indefinite matrix is avoided in the present work by introducing an artificial compressibility term in the continuity equation. Unequal order interpolation functions are used for pressure, velocity, temperature and concentration variables. The coefficient matrices are calculated using interpolation functions. Assembling of triangular elements in the discretized domain is discussed elaborately. The process of calculating boundary integrals is also discussed. The Newton-Raphson iteration technique along with Euler-backward scheme is used to solve the global matrix. The sample results are obtained for the convective flow of nanofluids in a concentric annulus. It shows that the annulus of having higher thickness is the best performer enhancing convective heat transfer rates.

General fully implicit discretization of the diffusion term for the finite volume method

ABSTRACTIn this paper, a fully implicit method for the discretization of the diffusion term is presented in the context of the cell-centered finite volume method. The newly developed fully implicit method is denoted by the modified implicit nonlinear diffusion (MIND) scheme. The method is used to solve several isotropic and anisotropic diffusion problems in two-dimensional domains covered with structured (quadrilateral elements) and unstructured (triangular elements) grid systems. The comparison of generated results with similar ones obtained using the semi-implicit scheme demonstrates the superior robustness and accuracy of the MIND scheme and its good convergence characteristics for all types of meshes.

Exact integration scheme for planewave-enriched partition of unity finite element method to solve the Helmholtz problem

In this paper, we present an exact integration scheme to compute highly oscillatory integrals that appear in the solution of the two-dimensional Helmholtz problem using the planewave-enriched partition of unity finite element method. In the proposed scheme, such oscillatory integrals are computed by a recursive application of the divergence theorem, eventually expressing the integrals in terms of evaluations of the corresponding integrands at the nodes of the finite element mesh. The number of such function evaluations is independent of the wave number k, which permits the scheme to be used for arbitrary high values of k. We consider finite element meshes with unstructured triangular and structured rectangular elements, and present numerical results for three canonical benchmark Helmholtz problems to demonstrate the accuracy and efficacy of the method.

Sensitivity Analysis of the Galerkin Finite Element Method Neutron Diffusion Solver to the Shape of the Elements

The purpose of the present study is the presentation of the appropriate element and shape function in the solution of the neutron diffusion equation in two-dimensional (2D) geometries. To this end, the multigroup neutron diffusion equation is solved using the Galerkin finite element method in both rectangular and hexagonal reactor cores. The spatial discretization of the equation is performed using unstructured triangular and quadrilateral finite elements. Calculations are performed using both linear and quadratic approximations of shape function in the Galerkin finite element method, based on which results are compared. Using the power iteration method, the neutron flux distributions with the corresponding eigenvalue are obtained. The results are then validated against the valid results for IAEA-2D and BIBLIS-2D benchmark problems. To investigate the dependency of the results to the type and number of the elements, and shape function order, a sensitivity analysis of the calculations to the mentioned parameters is performed. It is shown that the triangular elements and second order of the shape function in each element give the best results in comparison to the other states.

MARE2DEM: a 2-D inversion code for controlled-source electromagnetic and magnetotelluric data

This work presents MARE2DEM, a freely available code for 2-D anisotropic inversion of magnetotelluric (MT) data and frequency-domain controlled-source electromagnetic (CSEM) data from onshore and offshore surveys. MARE2DEM parametrizes the inverse model using a grid of arbitrarily shaped polygons, where unstructured triangular or quadrilateral grids are typically used due to their ease of construction. Unstructured grids provide significantly more geometric flexibility and parameter efficiency than the structured rectangular grids commonly used by most other inversion codes. Transmitter and receiver components located on topographic slopes can be tilted parallel to the boundary so that the simulated electromagnetic fields accurately reproduce the real survey geometry. The forward solution is implemented with a goal-oriented adaptive finite-element method that automatically generates and refines unstructured triangular element grids that conform to the inversion parameter grid, ensuring accurate responses as the model conductivity changes. This dual-grid approach is significantly more efficient than the conventional use of a single grid for both the forward and inverse meshes since the more detailed finite-element meshes required for accurate responses do not increase the memory requirements of the inverse problem. Forward solutions are computed in parallel with a highly efficient scaling by partitioning the data into smaller independent modeling tasks consisting of subsets of the input frequencies, transmitters and receivers. Non-linear inversion is carried out with a new Occam inversion approach that requires fewer forward calls. Dense matrix operations are optimized for memory and parallel scalability using the ScaLAPACK parallel library. Free parameters can be bounded using a new non-linear transformation that leaves the transformed parameters nearly the same as the original parameters within the bounds, thereby reducing non-linear smoothing effects. Data balancing normalization weights for the joint inversion of two or more data sets encourages the inversion to fit each data type equally well. A synthetic joint inversion of marine CSEM and MT data illustrates the algorithm's performance and parallel scaling on up to 480 processing cores. CSEM inversion of data from the Middle America Trench offshore Nicaragua demonstrates a real world application. The source code and MATLAB interface tools are freely available at http://mare2dem.ucsd.edu.

Direct numerical simulation for flow and heat transfer through random open-cell solid foams: Development of an IBM based CFD model

In the present contribution a sharp interface Immersed Boundary Method (IBM) for flow and heat transfer has been developed to fully resolve highly complex random solid structure on a non-body fitted Cartesian computational grid. A 3D image data set from a Micro-CT (computed tomography) scan of an actual foam geometry is usually converted into a surface mesh of unstructured triangular elements and the current frame work can embed it as an immersed boundary in the CFD domain. A second-order implicit (semi-implicit) method is used to incorporate fluid-solid coupling for flow (heat transfer) at the non-conforming immersed boundary in a structured grid. Both temperature Dirichlet and Neumann boundary conditions, as well as a conjugate heat transfer condition are incorporated at the fluid–solid interface. The proposed method is validated rigorously with existing literature results. The use of a Cartesian grid for flow makes this method robust, computational friendly, and at the same time it avoids the tedious volumetric mesh generation process for very complex shapes. To demonstrate the capability of the current method, it is applied to study flow and heat transfer of a typical foam sample with different flow properties. A closure for pressure drop and heat transfer coefficient are derived for the typical foam sample.

A local anisotropic adaptive algorithm for the solution of low-Mach transient combustion problems

A novel numerical algorithm for the simulation of transient combustion problems at low Mach and moderately high Reynolds numbers is presented. These problems are often characterized by the existence of a large disparity of length and time scales, resulting in the development of directional flow features, such as slender jets, boundary layers, mixing layers, or flame fronts. This makes local anisotropic adaptive techniques quite advantageous computationally. In this work we propose a local anisotropic refinement algorithm using, for the spatial discretization, unstructured triangular elements in a finite element framework. For the time integration, the problem is formulated in the context of semi-Lagrangian schemes, introducing the semi-Lagrange-Galerkin (SLG) technique as a better alternative to the classical semi-Lagrangian (SL) interpolation. The good performance of the numerical algorithm is illustrated by solving a canonical laminar combustion problem: the flame/vortex interaction. First, a premixed methane-air flame/vortex interaction with simplified transport and chemistry description (Test I) is considered. Results are found to be in excellent agreement with those in the literature, proving the superior performance of the SLG scheme when compared with the classical SL technique, and the advantage of using anisotropic adaptation instead of uniform meshes or isotropic mesh refinement. As a more realistic example, we then conduct simulations of non-premixed hydrogen-air flame/vortex interactions (Test II) using a more complex combustion model which involves state-of-the-art transport and chemical kinetics. In addition to the analysis of the numerical features, this second example allows us to perform a satisfactory comparison with experimental visualizations taken from the literature.

Galerkin and Generalized Least Squares finite element: A comparative study for multi-group diffusion solvers

In this paper, the solution of multi-group neutron/adjoint equation using Finite Element Method (FEM) for hexagonal and rectangular reactor cores is reported. The spatial discretization of the neutron diffusion equation is performed based on two different Finite Element Methods (FEMs) using unstructured triangular elements generated by Gambit software. Calculations are performed using Galerkin and Generalized Least Squares FEMs; based on which results are compared. Using the power iteration method for the neutron and adjoint calculations, the neutron and adjoint flux distributions with the corresponding eigenvalues are obtained. The results are then validated against the valid results for the IAEA-2D andBIBLIS-2D benchmark problems. The results of GFEM-2D (developed based on Galerkin FEM) and GELES-2D (developed based on Generalized Least Squares FEM) computer codes are also compared with results obtained from DONJON4 computer code. To investigate the validation of developed computer codes for the calculation with more than two energy groups, the calculations are performed for a benchmark problem with seven energy groups. To investigate the dependency of the results to the number of elements, a sensitivity analysis of the calculations to the number of elements is performed.