Biquadratic Elements Research Articles

Analysis of nonlinear problems using the Dual Mesh Control Domain Method with arbitrary meshes

The Dual Mesh Control Domain Method (DMCDM), proposed and developed by the second author, is a recent advancement in the field of numerical methods. This method represents a blend of the Finite Element Method (FEM) and the Finite Volume Method (FVM), each having its primary domains of popularity. The DMCDM incorporates the concepts of interpolation, duality, and numerical integration from the former and the idea of satisfying the integral (global) form of the differential (or balance) equations from the latter. In this paper, we present a fresh and systematic approach to solving two-dimensional nonlinear problems, including advection–diffusion problems, with arbitrary primal meshes. We develop an approach where element-level equations are derived and assembled in a manner similar to that of the FEM, as opposed to using nodal equations, as has been the convention in all of the previous works involving the DMCDM. Additionally, we extend the previous works to include higher-order interpolation, specifically quadratic interpolation. In short, we provide formulations for four-node bilinear and nine-node biquadratic elements of quadrilateral shape, enabling the application of the DMCDM to arbitrary primal and dual meshes. Finally, we explore the Newton iteration method for linearizing the nonlinear system of equations that arises from the DMCDM discretization. We present numerical results and comparisons against available exact and numerical solutions to verify the accuracy and robustness of the DMCDM.

Biquadratic element discrete duality finite volume method for solving elliptic equations on quadrilateral mesh

Nowadays numerical methods for handling partial differential equations is so vast that it is almost taken for granted. In this paper, we propose a discrete duality finite volume scheme with biquadratic elements and its application in solving two-dimensional elliptic equation on quadrilateral meshes. To test the robustness and efficiency of this method, we investigate several examples from physical and engineering fields with different parameters and coefficients. Numerical experiments, encompassing linear and nonlinear elliptic equations with constant or variable coefficients, reveal that the new method achieves optimal convergence in the continuous H1-norm, with a 2nd-order convergence, and in the L2-norm, with a 3rd-order convergence. Moreover, comparison with the existing biquadratic element finite volume scheme shows that the above method has an advantage of accuracy in the discrete L∞-norm on several distorted meshes.

Role of metal inserts for ‘targeted’, ‘uniform’ or ‘controlled’ microwave heating objectives involving generalized sets of dielectrics (Group 1 – 4)

Microwave heating occurs volumetrically leading to faster processing than conventional heating processes. Over the last few decades, the potential of microwave heating has been investigated for various thermal processing applications such as food processing, material synthesis, fuel production and waste treatment/remediation. Current work investigates the role of the metal inserts towards targeted, uniform and/or controlled microwave heating for the four groups of dielectrics (Group 1 – 4) involving lateral and isotropic microwave incidences. The microwave induced heat generation and evolving temperature distributions within the materials in the absence and presence of the metal insert are numerically determined via the mathematical models comprising of Helmholtz equation, energy balance equation and an integro-differential boundary condition representing the scattering of the microwave at the air-material interface. Galerkin finite element method with biquadratic elements along with Crank-Nicolson time integration scheme has been employed to solve the governing equations. It has been found that the metal insert can suitably alter the heating characteristics leading to either faster heating, better heating uniformity or more intense targeted heating of the materials. Uniform microwave heating is generally difficult to attain except for very small samples. This work shows that the metal inserts can significantly reduce the heating inhomogeneity of the moderately large samples involving Group 1 to Group 3 materials. In addition, the targeted heating effect of the materials (Groups 1 to 4) can also be intensified by the use of the metal insert, as has been illustrated for various test cases. Especially, the metal insert can lead to targeted heating at the incident surface irrespective of the material or sample dimension, and that is difficult to attain except for the large samples. The metal inserts are also found to be efficient to control the rapid microwave heating experienced by the small samples of lossy materials (Group 1 and 3 materials) or accelerate the slower heating of the large samples resulting in significant energy savings, especially for Group 1 to Group 3 materials.

Irreversibility analysis for the natural convection of Casson fluid in an inclined porous cavity under the effects of magnetic field and viscous dissipation

The non-Newtonian fluids are increasingly being employed in various engineering and industrial processes. The Casson fluid model is also used to characterize the non-Newtonian fluid behavior and has great importance in polymer processing industries and biomechanics. Motivated by these developments, the present study explores the entropy generation for the natural convection of Casson fluid in a porous, partially heated square enclosure considering the effects of horizontal magnetic field, cavity inclination and viscous dissipation. The bottom wall of the enclosure is considered partially heated, whereas the left and right walls are taken as cold at constant temperature. The top wall and the remaining portions of the bottom wall are adiabatic. The dimensionless form of the governing conservation laws is simulated via higher order Galerkin finite element method. Particularly, the discretization is performed using biquadratic elements for velocity and temperature components whereas the discontinuous linear elements employed for the pressure. The discretized nonlinear systems are handled by implementing the adaptive Newton-multigrid solver. In order to increase the reliability of the computed results, the designed solver is validated qualitatively and quantitatively for the available numerical and experimental data. The simulated results are analyzed through isotherms, streamlines and two dimensional plots. Moreover, the average entropy generation, kinetic energy and temperature are also computed and analyzed. The controlling parameters are Hartmann number (Ha=0−100), Darcy number (Da=10−4), Prandtl number (Pr=7), Rayleigh number (Ra=105), Casson fluid parameter (γ=0.1−10), cavity inclination (ϕ=0°−90°) and Eckert number (Ec=10−6−10−4). It is concluded that the average heat transfer rate decreases whereas the total entropy generation increases by increasing the cavity inclination (ϕ) for each value of Ha. Further, the irreversibilities due to heat transfer and magnetic field both are increasing for function of ϕ.

Numerical Analysis of Least-Squares Group Finite Element Method for Coupled Burgers' Problem

In this paper, a least squares group finite element method for solving coupled Burgers' problem in 2-D is presented. A fully discrete formulation of least squares finite element method is analyzed, the backward-Euler scheme for the time variable is considered, the discretization with respect to space variable is applied as biquadratic quadrangular elements with nine nodes for each element. The continuity, ellipticity, stability condition and error estimate of least squares group finite element method are proved. The theoretical results show that the error estimate of this method is . The numerical results are compared with the exact solution and other available literature when the convection-dominated case to illustrate the efficiency of the proposed method that are solved through implementation in MATLAB R2018a.

Monotonicity correction for second order element finite volume methods of anisotropic diffusion problems

We apply the monotonicity correction to second-order element finite volume methods, which include quadratic element on triangular meshes and biquadratic element on quadrilateral meshes, for anisotropic diffusion problems, and obtain a second-order monotone finite volume scheme. The main ideas of monotonicity correction are as follows: When we formulate the second-order element finite volume schemes, we need to calculate line integrals on the boundary line segment of the dual element, and regard these line integrals as numerical fluxes, which are in the form of multi-point flux. Among these numerical fluxes, we can separate a two-point flux structure on both sides of the line segment. Therefore, we can apply nonlinear monotone correction to these numerical fluxes, then we obtain a corrected second order element finite volume schemes whose stiffness matrix is monotone matrix, so the first high order unconditional positivity-preserving finite volume schemes are obtained by us. Moreover, we analyze the truncation error of the corrected numerical fluxes. Furthermore, for the time dependent problem, we also apply monotone correction to time derivative term. Numerical experiments show that the corrected second order element finite volume schemes have monotonicity, and maintain the convergence order of the original schemes, which are second order in H1 norm and third order in L2 norm.

Study of fluid flow inside closed cavities using computational numerical methods

The temperature distribution and distortion of fluid flow inside the closed cavities, square and triangle, are studied for different boundary conditions. Two different conditions of thermal boundary conditions are used for studying square cavities: (i) Left wall is hot, right wall is cold, top and bottom walls are adiabatic. (ii) Left and right walls are cold, top wall is adiabatic, bottom wall is hot. For triangular enclosure, the boundary conditions are (i) the vertical wall is insulated, bottom wall is hot. (ii) The vertical wall is hot, the bottom wall insulated and the inclined walls are kept cold in both conditions. The velocity of the flow is observed by means of stream function and the temperature distribution is displayed in the form of contours. The study is carried out in ANSYS software. The mathematical procedure for solving the nonlinear system of partial differential equations by penalty finite element method involving bi-quadratic elements is also discussed in detail.

ГРАНИЧНО-ЭЛЕМЕНТНЫЙ АНАЛИЗ РАСПРОСТРАНЕНИЯ ВОЛН В ПОРОВЯЗКОУПРУГОМ СЛОИСТОМ ПОЛУПРОСТРАНСТВЕ И ПОЛУПРОСТРАНСТВЕ С ПОЛОСТЬЮ

The paper is dedicated to the wave propagation a porous-viscoelastic material. As a mathematical model of a fully saturated poroelastic medium, we consider the Biot model with four basic functions – pore pressure and skeleton movements. The Biot model is supplemented by the principle of elastic and viscoelastic reaction correspondence. The skeleton of a porous material is assumed to be viscoelastic material. A model of a standard viscoelastic solid is spplied to describe the viscoelastic properties of a skeleton. The initial boundary-value problem is reduced to a boundary-value problem by formal application of the Laplace transform. To solve boundary integral equations, the boundary element method is performed. Quadrangular eight-node biquadratic elements are used for boundary element discretization. Numerical integration is carried out according to Gaussian quadrature formulas using algorithms for lowering the order and eliminating features. To obtain a solution in explicit time, numerical inversion of the Laplace transform is applied based on the Durbin algorithm with a variable frequency step. This study is a development of the existing boundary-element technique for solving problems on layered porous-elastic half-spaces. This will allow you to take into account the heterogeneity of the soil in depth. The problem of the action of a vertical force in the form of the Heaviside function on the surface of a layered porous-elastic half-space and a half-space with a cavity is considered. Variants of a homogeneous and heterogeneous half-space are considered. Under the model of heterogeneity we understand the piecewise homogeneous solid. The responses of the boundary displacements on the surface of the half-space are presented. The effect of the viscoelastic material model parameter on the dynamic response of displacements is demonstrated. It is established that the viscosity parameters have a significant effect on the nature of the distribution of parameters of wave processes.

ФІЗИЧНО АДЕКВАТНА КОНДЕНСАЦІЯ І МІШАНІ МОДЕЛІ СЕРЕНДИПОВИХ ЕЛЕМЕНТІВ

У роботі розглядається серендипова версія квадратично-кубічної інтерполяції на канонічному квадраті (|x| ≤ 1, |y| ≤ 1). У напрямку вісі 0x функція змінюється за законом кубічної параболи, у напрямку 0y – за законом квадратичної параболи. Лагранжевий прообраз такого елемента має 12 вузлів (два внутрішніх). Як відомо, небажані внутрішні вузли виключають, щоб отримати серендипову модель. Традиційна процедура конденсації (редукції) полягає у складанні і розв’язуванні СЛАР з матрицею 12×12. Далі, щоб усунути внутрішні вузли, потрібно знайти "рецепт" конденсації, тобто побудувати лінійну залежність внутрішніх параметрів (двох) від граничних (десяти). Відомі приклади свідчать, що математично обґрунтований "рецепт" конденсації не гарантує фізичної адекватності спектра вузлових навантажень серендипових моделей. Так було з біквадратичним елементом ("рецепт" Джордана, 1970) і трикутником третього порядку ("рецепт" Сьярле-Равьяра, 1972). Щоб уникнути аномалій в спектрі вузлових навантажень, потрібно починати з побудови бажаного спектра. Це обернена задача, коли спочатку вибирають бажані інтегральні характеристики, а після цього визначають базис, який реалізує ці характеристики. Саме такий "нематричний" підхід запропоновано в роботі. Важлива властивість нематричної редукції полягає в тому, що вона виключає внутрішні вузли, але зберігає внутрішні параметри. Наявність "прихованих" параметрів дозволяє керувати формоутворенням альтернативних серендипових поверхонь.

Finite element simulations for stationary Bingham fluid flow past a circular cylinder

This paper presents the stationary Bingham fluid flow simulations past a circular cylinder placed in a channel. The governing equations of motion are discretized using the mixed finite element method. The biquadratic element Q2 is used to approximate the velocity space and Pdisc for the pressure space. This LBB-stable finite element pair $$Q_{2} /P_{1}^{\text{disc}}$$ leads to an accuracy of third and second order in L2-norm for velocity and pressure approximations, respectively. The discrete version of the nonlinear system is linearized by the Newton’s method, and the linear subproblems as an inner loop are coped with UMFPACK solver. Code validation and grid independence study is also presented to confirm the reliability of the results. Simulations are carried out for various values of dimensionless Bingham number Bn, and its impact is investigated with great detail by demonstrating the velocity, viscosity and pressure contours. The benchmark quantities like drag coefficient (CD) and lift coefficient (CL) are also discussed graphically and quantitatively. Moreover, two different locations of the circular cylinder are considered for the analysis of hydrodynamic forces.

Role of finite element based grids and simulations on evaluation of Nusselt numbers for heatfunctions within square and triangular cavities involving multiple discrete heaters

Nusselt number is an important non-dimensional parameter which quantifies the heat transfer rate. Local Nusselt number is useful in predicting the heat transfer rate along the various hot and cold sections of the side walls in a discretely heated enclosed cavity. In addition, the overall heat balance in an enclosed cavity (total heat delivered by the hot isothermal walls should be equal to the total heat gained by the cold isothermal walls) can be validated via the average Nusselt numbers. Current finite element based simulations and post-processing have been carried out in order to analyze the influence of the multiple heaters on the Nusselt number along various sections (hot and cold) of the side walls in discretely heated square and triangular (design 1 and design 2) cavities. The working fluid is considered to be air (Pr=0.7) and the numerical studies have been carried out for a large range of Rayleigh number (Ra=103–105) for four different biquadratic elements (24×24, 28×28, 32×32 and 34×34). The current work also estimates the fractional error in the heat balance (ϵ) and it is clearly inferred that ϵ is comparatively lower for 34×34 biquadratic elements. Current work also reveals that the fractional error (ϵ) is mainly induced due to the sharp variations in the Nusselt number at the cold-hot junctions along the side walls. The present study also involves the detailed evaluation of the heatfunction (Π) expressions along the cold-hot junctions of the side walls. The computations of the heatfunctions are intrinsically related to the Nusselt numbers of the hot-cold junctions.

Modeling of Wave Propagation in the Unsaturated Soils Using Boundary Element Method

Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.

On the finite element based evaluation of Nusselt numbers for curved walls

Two heating strategies are considered such as (a) differential heating with the curved side walls and (b) Rayleigh–Bénard heating with the curved horizontal walls for the study of natural convection. The simulations are carried out involving various Rayleigh numbers (103 ≤ Ra ≤ 105) and Prandtl number, Pr=7 for various elements with different cases based on the wall concavities and convexities. The Gaussian quadrature based Galerkin finite element method has been used to obtain the smooth solutions of velocity and temperature components within the curved computational domain. The local Nusselt numbers along the curved walls are derived in a generic form for various concave and convex walls and the local Nusselt numbers are evaluated based on the finite element method using the element basis functions. The error in the heat balance for the differential heating case is larger than the Rayleigh–Bénard heating strategy for all the concave and convex enclosures. At a specific heating strategy with a given element number, the error is larger for the convex enclosure with the high wall curvature. In addition, the minimum error in the heat balance is obtained for the finest grid with 32×32 bi-quadratic elements involving both the heating strategies with the concave and convex walls. The convergence in the local and average Nusselt numbers via the finite element method is obtained even at the coarser grid with 16×16 bi-quadratic elements corresponding to 33×33 grid points for most of the cases. The comparative study indicates that, the finite volume method [1] and finite difference method [2] require 121×121 grid points to obtain the converged solution whereas, the finite element method offers the converged solution for the grid corresponding to 33×33 grid points for natural convection in curved/wavy domains.

Pricing swing options in electricity markets with two stochastic factors using a partial differential equation approach

In this paper, we consider the numerical valuation of swing options in electricity markets based on a two-factor model. These kinds of contracts are modeled as path dependent options with multiple exercise rights. From a mathematical point of view, the valuation of these products is posed as a sequence of free boundary problems, where two exercise rights are separated by a time period. In order to solve the pricing problem, we propose appropriate numerical methods based on a Crank-Nicolson semi-Lagrangian method combined with biquadratic Lagrange finite elements for the discretization of the partial differential equation. In addition, we use an augmented Lagrangian active set method to cope with the early exercise feature when it appears. Moreover, we derive appropriate artificial boundary conditions to treat the unbounded domain numerically. Finally, we present some numerical results to illustrate the proper behavior of the numerical schemes.

Error Analysis of a Dual-parametric Bi-quadratic FEM in Cavitation Computation in Elasticity

The orientation-preservation conditions and approximation errors of a dual-parametric bi-quadratic finite element method for the computation of both radially symmetric and general nonsymmetric cavity solutions in nonlinear elasticity are analyzed. The analytical results allow us to establish, based on an error equidistribution principle, an optimal meshing strategy for the method in cavitation computation. Numerical results are in good agreement with the analytical results.

Boundary Element Method with Runge-Kutta Convolution Quadrature for Three-Dimensional Dynamic Poroelasticity

The paper contains a brief introduction to the state of the art in poroelasticity models, in BIE & BEM methods application to solve dynamic problems in Laplace domain. Convolution Quadrature Method is formulated, as well as Runge-Kutta convolution quadrature modification and scheme with a key based on the highly oscillatory quadrature principles. Several approaches to Laplace transform inversion, including based on traditional Euler stepping scheme and Runge-Kutta stepping schemes, are numerically compared. A BIE system of direct approach in Laplace domain is used together with the discretization technique based on the collocation method. The boundary is discretized with the quadrilateral 8-node biquadratic elements. Generalized boundary functions are approximated with the help of the Goldshteyn’s displacement-stress matched model. The time-stepping scheme can rely on the application of convolution theorem as well as integration theorem. By means of the developed software the following 3d poroelastodynamic problem were numerically treated: a Heaviside-shaped longitudinal load acting on the face of a column.

Effect of Inclination Angle of a Rhombic Enclosure on Natural Convection due to Differential Heating and Rayleigh-Bѐnard Configuration

This article analyzes the detailed natural convection phenomena for a rhombic enclosure with differential heating and Rayleigh-Bѐnard configuration. A bi-quadratic element has been used to calculate the differential fluxes in the Navier-Stokes equations. A fourth order artificial viscosity was used to stabilize the numerical residue. The present numerical solution is performed over a wide range of parameters; 10^3 ≤ Ra ≤ 10^8, 15^0 ≤ ϕ ≤ 165^0 for differential heating and 10^3 ≤ Ra ≤ 10^6, 15^0 ≤ ϕ ≤ 90^0 for Rayleigh-Bѐnard configuration. The analysis of the net convective heat transfer across the mid-passage or the mid-height is used to identify the contribution of vortex motion from conduction dominate to convection dominate. The overshoots or undershoots of this net convective contribution is highly related to inclination angle of the rhombic enclosure and also the thermal boundary conditions. The compressibility effect slightly alters the overall performance and overshoots or undershoots of the net heat transfer by less than 1% value. Average Nusselt number distributions show that heat transfer rate is maximum for ϕ=90o in differential heating case, while for Rayleigh-Bѐnard convection, the heat transfer rate is maximum for ϕ=75o except for ϕ=15o at Rayleigh number 10^3 where conduction heat transfer is dominate. There is a linearity between the average Nusselt number and log(Ra) for all the inclination angles for both cases. Results of the study shows that the slope of the linearity is steeper for smaller or wider inclination angles when convective heat transfer is dominate i.e., at larger Rayleigh number.

Error Analysis for Constrained First-Order System Least-Squares Finite-Element Methods

In this paper, a general error analysis is provided for finite-element discretizations of partial differential equations in a saddle-point form with divergence constraint. In particular, this extends upon the work of [J. H. Adler and P. S. Vassilevski, Springer Proc. Math. Statist. 45, Springer, New York, 2013, pp. 1--19], giving a general error estimate for finite-element problems augmented with a divergence constraint and showing that these estimates are obtained for problems such as diffusion and Stokes' using the first-order system least-squares (FOSLS) finite-element method. The main result is that by enforcing the constraint on a ${\mathbf H}^1$-equivalent FOSLS formulation one maintains optimal convergence of the FOSLS functional (i.e., the energy norm of the error) while guaranteeing the conservation of the divergence constraint (i.e., mass conservation in some examples). The error estimates and results depend on using finite elements for the constraint space that are inf-sup stable when paired with the spaces used for the original unknowns. This includes using discontinuous spaces on coarse meshes and pairing with standard bilinear or biquadratic elements in order to confirm the results.

Analysis of heatlines during natural convection within porous square enclosures: Effects of thermal aspect ratio and thermal boundary conditions

Numerical investigation of natural convection within porous square enclosures has been performed for various thermal boundary conditions based on thermal aspect ratio on bottom and side walls. Penalty finite element analysis with bi-quadratic elements is used to solve the governing equations. The numerical solutions are studied in terms of streamlines, isotherms, heatlines, local and average Nusselt numbers for a wide range of parameters Da(10−5–101), Pr(0.015–1000) and Ra(Ra=103–105). At low Darcy number (Da=10−5), heatlines are perpendicular to the isotherms indicating conduction dominant heat transfer. As Da increases to 10−3, convection is initiated and the thermal mixing has been observed at the central regime for all As. At low Prandtl number (Pr=0.015) with high Darcy number (Da=10−2 and Da=101), multiple circulations are observed in streamlines and heatlines and they suppressed for higher Prandtl number (Pr=1000). Isotherms are highly compressed along bottom wall at higher Prandtl numbers (Pr=0.7 and 1000) at A=0.1 and 0.5. Temperature gradient is found to be high at the center of the bottom wall for A=0.1 due to dense heatlines at that zone and that decreases as A increases from 0.1 to 0.9, irrespective of Pr, Da. Also, the temperature gradient is smaller at the top portion of side walls for A=0.1 due to sparse heatlines along those zones and that is high for A=0.9 due to dense heatlines. Distribution of heatlines illustrate that significant heat transport occurs from hot bottom wall to the top portion side walls at higher Darcy number (Da=101). It is found that Nub attains maximum at X=0.5 and minimum at corners for Da=10−5, whereas that exhibits sinusoidal variation for Da=10−3 and Da=101 irrespective of Pr and A. It is also found that Nul follows wavy pattern at low Prandtl number (Pr=0.015) with higher Darcy number (Da=101) irrespective of A due to larger gradients of heatfunctions at several locations of left wall. The average Nusselt number show that the overall heat transfer rate is high at A=0.1 compared to that of A=0.5 and A=0.9 irrespective of Da and Pr due to larger gradients of heatfunctions at A=0.1.

Higher‐order extended FEM for weak discontinuities – level set representation, quadrature and application to magneto‐mechanical problems

SUMMARYIn this article, we present the application of bilinear and biquadratic extended FEM (XFEM) formulations to model weak discontinuities in magnetic and coupled magneto‐mechanical boundary value problems. For properly resolving the location of curved interfaces and the discontinuous physical behaviour, the major part of the contribution is devoted to review and develop methods for level set representation of curved interfaces and numerical integration of the weak form in higher‐order XFEM formulations. In order to reduce the complexity of the representation of curved interfaces, an element local approach that allows for an automated computation of the level set values and also improves the compatibility between the level set representation and the integration subdomains is proposed. Integration rules for polygons and strain smoothing are applied in conjunction with biquadratic elements and compared with curved integration subdomains. Eventually, a coupled magneto‐mechanical demonstration problem is modelled and solved by XFEM. For demonstration purposes, a magneto‐mechanical coupling due to magnetic stresses is considered. Errors and convergence rates are analysed for the different level set representations and numerical integration procedures as well as their dependence on the ratio of material parameters at an interface. Copyright © 2012 John Wiley & Sons, Ltd.