Linear Quadrilateral Elements Research Articles

Verification of a High-Order FEM-based CFD Code using the Method of Manufactured Solutions

A high-order computational fluid dynamics (CFD) code capable of solving compressible turbulent flow problems was developed. The CFD code employs the Flowfield Dependent Variation (FDV) scheme implemented in a Finite Element Method (FEM) framework. The FDV scheme is basically derived from the Lax-Wendroff Scheme (LWS) involving the replacement of LWS’s explicit time derivatives with a weighted combination of explicit and implicit time derivatives. The code utilizes linear, quadratic and cubic isoparametric quadrilateral and hexahedral Lagrange finite elements with corresponding piecewise shape functions that have formal spatial accuracy of second-order, third-order and fourth-order, respectively. In this paper, the results of observed order-of-accuracy of the implemented FDV FEM-based CFD code involving grid and polynomial order refinements on uniform Cartesian grids are reported. The Method of Manufactured Solutions (MMS) is applied to governing 2-D Euler and Navier-Stokes equations for flow cases spanning both subsonic and supersonic flow regimes. Global discretization error analyses using discrete 𝐿2 norm show that the spatial order-of-accuracy of the FDV FEM-based CFD code converges to the shape function polynomial order plus one, in excellent agreement with theory. Uniquely, this procedure establishes the wider applicability of MMS in verifying the spatial accuracy of a high-order CFD code.

Analytical expressions for singular integrals arising from the 3D Laplace and Stokes kernels when using constant or linear triangular and quadrilateral boundary elements

In this paper we present analytical expressions for the computation of singular integrals obtained in the discretisation of boundary integral equations for the Laplace and creeping flow (Stokes) problems with triangular or quadrilateral boundary elements with linear interpolation of the potential and constant interpolation of the flux. We compare the singular integrals computed with the presented analytical expressions with the same integrals computed with numerical quadrature and find that a considerably larger computational effort has to be made for numerical quadrature to achieve high accuracy than with the analytical expression. Furthermore, we show that the accuracy in solving a Laplace test case and a creeping flow test case using analytical expressions for singular integrals is better than the accuracy achieved with numerical quadrature. The analytical expressions are listed in the appendix of the paper and their implementation in computer code is available online.

Non-conforming and conforming five-node quadrilateral graded finite elements

This study evaluates the performance of conforming and non-conforming transition graded finite elements dealing with graded materials. Two-dimensional 5-node non-conforming quadrilateral element (Q5) is used to transition from one large 4-node linear quadrilateral to two small 4-node linear quadrilateral elements (Q4). In addition, a conforming Q5 element is used in between one Q4 and one 8-node quadratic elements (Q8). A single element test of Q5 graded element is done. Then, the accuracy and convergence study of Q5 graded element are examined. We found that the use of Q5 transition elements improved computational efficiency while maintaining the accuracy of the solutions.

A linear quadrilateral shell element for laminated composites

A linear quadrilateral shell element for laminated composites

Numerical analysis of the magnetic shape memory effect based on the absolute nodal coordinate formulation

In this paper, the absolute nodal coordinate formulation (ANCF) is applied to simulate the magnetic shape memory effect. Using the absolute nodal coordinate formulation makes it possible to describe complicated or large deformation cases. The nonlinear bidirectional coupling terms between the mechanical and magnetic field are taken into account in the analysis of the single-crystalline Ni-Mn-Ga sample. A two-loop iteration procedure with variable steps is implemented to predict the magnetic-field-induced strain (MFIS) in the specimen under a changing external magnetic field and a constant auxiliary compression. In addition, the proposed approach is used to track the superelastic behavior of the magnetic shape memory alloy when subjected to a constant magnetic field. The approach effectively describes the hysteresis and superelastic phenomenon of the shape memory effect. The solution is compared here with solutions obtained using classical linear and quadratic quadrilateral elements. The deviation observed in the solution is discussed, and its cause is further clarified from a two-domain pure magnetostatic analysis of a permanent magnet. It is found that the accurate solution of such problems is associated with discontinuity of the normal component of the magnetic potential gradient across the domain interface. Special measures must be taken to make the absolute nodal coordinate formulation element compatible with the discontinuity. A mixed FEs strategy, which adopts ANCF FE in the displacement field solver and classical FE in the magnetic field solver, is proposed as an alternative option to rectify the problem, which is verified by predicting the MFIS and the dynamic mechanical response of a sample under cyclic compression.

An explicit coupled MPM formulation to simulate penetration problems in soils using quadrilateral elements

This paper describes an MPM formulation using linear quadrilateral elements suitable for soil-structure-interaction problems. The volumetric locking and stress-oscillations are mitigated using a reduced integration technique and the Gauss integration scheme, respectively. The formulation can be used with both structured and unstructured computational meshes. In addition, an improved calculation scheme is proposed to obtain accurate contact reaction forces, especially for contacts between non-porous structures and soils with high liquid pressures. The formulation is validated by simulating a wide range of applications such as a water dam break, 1D large-strain consolidation, and the bearing capacity of a strip footing. Finally, the installation of a cone penetrometer in a centrifuge is successfully simulated in soft soils and compared with the experimental data for the entire range of drainage conditions.

Optimal local truncation error method for solution of elasticity problems for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes

The optimal local truncation error method (OLTEM) with unfitted Cartesian meshes was recently developed for PDEs with homogeneous materials on regular and irregular domains as well as for the scalar time-dependent wave and heat equations for heterogeneous materials with irregular interfaces. Here, OLTEM is extended to a system of time-independent elastic PDEs for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes. We show the development of OLTEM for the 2D elasticity equations using compact 9-point stencils that are similar to those for linear quadrilateral finite elements. The interface conditions on the interfaces where the jumps in material properties occur are added to the expression for the local truncation error and do not change the width of the stencils. There are no unknowns on interfaces between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal second order of accuracy for OLTEM with 9-point stencils on unfitted Cartesian meshes. Numerical experiments for elastic heterogeneous materials with irregular interfaces show that at the same number of degrees of freedom: a) OLTEM with unfitted Cartesian meshes is more accurate than linear finite elements with similar stencils and conformed meshes; b) up to engineering accuracy of OLTEM with unfitted Cartesian meshes is even more computationally efficient than quadratic and cubic finite elements with much wider stencils and conformed meshes. The proposed technique yields accurate numerical results for heterogeneous materials with big contrasts in the material properties of different components. Due to the computational efficiency and trivial unfitted Cartesian meshes that are independent of irregular geometry, the proposed technique does not require remeshing for the shape change of irregular geometry and it will be effective for many design and optimization problems.

Optimal local truncation error method for solution of wave and heat equations for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes

Recently we have developed the optimal local truncation error method (OLTEM) for PDEs with constant coefficients on irregular domains and unfitted Cartesian meshes. However, many important engineering applications include domains with different material properties (e.g., different inclusions, multi-material structural components, etc.) for which this technique cannot be directly applied. In the paper OLTEM is extended to a much more general case of PDEs with discontinuous coefficients and can treat the above-mentioned applications. We show the development of OLTEM for the 1-D and 2-D scalar wave equation as well as the heat equation using compact 3-point (in the 1-D case) and 9-point (in the 2-D case) stencils that are similar to those for linear quadrilateral finite elements. Trivial unfitted Cartesian meshes are used for OLTEM with complex interfaces between different materials. The interface conditions on the interfaces where the jumps in material properties occur are added to the expression for the local truncation error and do not change the width of the stencils. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique at a given width of stencil equations. In contrast to the second order of accuracy for linear finite elements, OLTEM provides the fourth order of accuracy in the 1-D case and in the 2-D case for horizontal interfaces as well as the third order of accuracy for the general geometry of smooth interfaces. The numerical results for the domains with complex smooth interfaces show that at the same number of degrees of freedom, OLTEM is even much more accurate than quadratic finite elements and yields the results close to those for cubic finite elements with much wider stencils. The wave and heat equations are uniformly treated with OLTEM. OLTEM can be directly applied to other partial differential equations.

Non-Hydrostatic Discontinuous/Continuous Galerkin Model for Wave Propagation, Breaking and Runup

This paper presents a new depth-integrated non-hydrostatic finite element model for simulating wave propagation, breaking and runup using a combination of discontinuous and continuous Galerkin methods. The formulation decomposes the depth-integrated non-hydrostatic equations into hydrostatic and non-hydrostatic parts. The hydrostatic part is solved with a discontinuous Galerkin finite element method to allow the simulation of discontinuous flows, wave breaking and runup. The non-hydrostatic part led to a Poisson type equation, where the non-hydrostatic pressure is solved using a continuous Galerkin method to allow the modeling of wave propagation and transformation. The model uses linear quadrilateral finite elements for horizontal velocities, water surface elevations and non-hydrostatic pressures approximations. A new slope limiter for quadrilateral elements is developed. The model is verified and validated by a series of analytical solutions and laboratory experiments.

Finite element approach for numerical integration over family of eight node linear quadrilateral element for solving Laplace equation

This paper describes the finite element solution of the boundary value problem or Laplace equation in curved and square region, the boundary region is divided into arbitrary triangles, and every triangle is divided in to three eight node quadrilateral elements, to generate a mesh over rectangular and curved region, performance of the new approach are demonstrated with numerical examples.

Multi-frequency sequential and simultaneous parametric shape optimization of reactive silencers

Silencers are widely used to attenuate unwanted noise emitted by engines, exhaust, and ventilation systems or from intermittent gas flow in pipes. In recent years, the study of noise control using optimization techniques applied to silencers has been growing. The main objective of the present work is to parametrically optimize the shape of a compact reactive silencer with a single expansion chamber, modifying its construction dimensions. Sequential and simultaneous optimizations are applied to the shapes of three reactive filter types so that their sound transmission losses are maximized over pre-established frequency ranges. The optimized reactive filters are the expansion chamber with modified surface profile, extended ducts and the bi-partition of the chamber. The optimization process is carried out with the aid of the finite element method using a formulation for axisymmetric problems and with the models being discretized with linear quadrilateral elements. The optimizations were evaluated in 16 frequency range combinations totaling 98 possible results. The results showed that both optimization techniques are efficient in improving the efficiency of silencers above the first TL drop frequency of the base model, which is composed of a single expansion chamber. However, the sequential strategy presented optimized models with more regular final shapes. In addition, the methodologies are validated using the traditional three-dimensional Finite Element Method and with the experimental verification of two optimized models.

Bending and free vibration behaviors of composite plates using the C0-HSDT based four-node element with in-plane rotations

In this paper the smoothed four-node element with in-plane rotations MISQ24 is combined with a C0-type higher-order shear deformation theory (C0-HSDT) to propose an improved linear quadrilateral plate element for static and free vibration analyses of laminated composite plates. This improvement results in two additional degrees of freedom at each node and require no shear correction factors while ensuring the high precision of numerical solutions. Composite plates with different lay-ups, boundary conditions and various geometries such as rectangular, skew and triangular plates are analyzed using the proposed element. The obtained numerical results are compared with those from previous studies in the literature to demonstrate the effectiveness, the reliability and the accuracy of the present element.
 Keywords:
 composite laminated plates; bending problems; free vibration; C0-HSDT; MISQ24.

Functional Reconstitution of Reissner’s Mixed Variational Theorem for Finite Element Applications

The functional reconstitution technique is used for the first time to theoretically and a priori assess the numerical performance of Reissner’s Mixed Variational Theorem in the case of linear quadrilateral finite element, Generalized Unified Formulation, and composite materials. From the finite element discretization, the semi-complementary energy is reconstructed in integral form. Analytical expressions for the exact and spurious terms are obtained. Using the elasticity solution for a simply supported plate, it is numerically shown that these spurious contributions do not significantly alter the quality of the approximation of the total level of energy, with overall excellent computational performance. On the other hand, when the starting functional is the strain energy and the principle of virtual displacement is adopted, the finite-element-introduced additional terms and in particular the contributions associated to the transverse shear stresses are demonstrated to add a very large error, which is a priori proven to be eliminated with selective integration. The presented technique could be extended to analyze the performance of other finite elements (e.g., higher-order triangular ones) within Reissner’s Mixed Variational Theorem and Generalized Unified Formulation.

Nodal integral method for convection-diffusion transport using linear and higher order quadrilateral elements

Generic nodal integral method (NIM) based scheme, utilizing nine noded 2D quadratic elements along with four noded 2D linear elements, is developed to solve the fluid flow and heat transfer equations in complex geometries. Non-linear (quadratic) quadrilateral elements are used for discretization of boundary region and linear quadrilateral elements are used for interior region. Lagrange interpolation functions are used to map both type of elements to corresponding square computational elements. The scheme for Neumann and mixed type of boundary conditions are also developed for quadratic elements. C1 type continuity condition is imposed at the interfaces of adjacent elements. Numerical results are compared with analytical solutions for diffusion and advection-diffusion equations. Results for Navier–Stokes equations in curved domain are compared with previously reported experimental as well as numerical results. The comparative study has also been done between presently developed scheme using quadratic and linear elements (referred as scheme-1) and scheme based on complete discretization with linear elements (referred as scheme-2). The comparison shows that both the schemes are of nearly second order accurate while the scheme-1 is more accurate than scheme-2. The results show that the efficient mapping of curved surface with quadratic elements improves the accuracy of NIM schemes.

Finite element analysis of lung ultrasound surface wave elastography

Interstitial lung disease manifested as fibrotic and stiffened lung tissue, resulting in symptoms such as dyspnea and may lead to respiratory failure. We recently developed a lung ultrasound surface wave elastography for evaluating the elastic properties of superficial lung tissue. In this abstract, we present a finite element analysis to investigate the effects of pleural effusion on the shear wave propagation of superficial lung tissue. The muscle, pleural effusion, and lung tissue were simulated by a 2D planar model of an infinite elastic medium with densities for different materials. A nearly incompressible, linear elastic model was used to model muscle, pleural effusion and lung. The model was excited using a line source on the muscle surface. Harmonic excitations were performed at 100, 150 and 200 Hz with duration of 0.1 s. The boundaries of lung and muscle were attached to an infinite. The mesh of muscle, pleural effusion, and lung were constructed using linear quadrilateral elements. The infinite region was meshed by infinite elements (type CINPE4). The dynamic responses of the tissue model to the excitations were solved by the ABAQUS explicit dynamic solver with automatic step size control. Insignificant difference in shear wave speeds of lung tissue at different excitation frequencies was found in both simulations with and without taking into account the pleural effusion, indicating that pleural effusion does not affect the measurements of shear wave speeds in ultrasound elastography.Interstitial lung disease manifested as fibrotic and stiffened lung tissue, resulting in symptoms such as dyspnea and may lead to respiratory failure. We recently developed a lung ultrasound surface wave elastography for evaluating the elastic properties of superficial lung tissue. In this abstract, we present a finite element analysis to investigate the effects of pleural effusion on the shear wave propagation of superficial lung tissue. The muscle, pleural effusion, and lung tissue were simulated by a 2D planar model of an infinite elastic medium with densities for different materials. A nearly incompressible, linear elastic model was used to model muscle, pleural effusion and lung. The model was excited using a line source on the muscle surface. Harmonic excitations were performed at 100, 150 and 200 Hz with duration of 0.1 s. The boundaries of lung and muscle were attached to an infinite. The mesh of muscle, pleural effusion, and lung were constructed using linear quadrilateral elements. The infinite ...

Finite element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for nine node linear convex quadrilateral of Lagrange family

This paper presents an explicit integration scheme to compute the stiffness matrix of a nine node linear convex quadrilateral element of Lagrange family using symbolic mathematics and discretisation of polygonal domain by such finite elements using a novel auto mesh generation technique , In finite element analysis, the boundary value problems governed by second order linear partial differential equations, the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+ ) in the bivariates over a nine node 2-square (-1 ) with nodes at the points {(-1,-1),(1,-1),(1,1),(-1,1),(0,-1),(1,0).(0,1),(- 1,0),(0,0)} in local parametric space. In this paper, we have computed these integrals in exact forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be applied to solve boundary value problems in continuum mechanics over convex polygonal domains. We have also developed a novel auto mesh generation technique of all nine node linear convex quadrilaterals for a polygonal domain which provides the nodal coordinates and element connectivity. We have used the explicit integration scheme and this novel auto mesh generation technique to solve the Poisson equation u ,where u is unknown physical variable and in with given Dirichlet boundary conditions over the given convex polygonal domain.

A comparison between different finite elements for elastic and aero-elastic analyses

In the present paper, a comparison between five different shell finite elements, including the Linear Triangular Element, Linear Quadrilateral Element, Linear Quadrilateral Element based on deformation modes, 8-node Quadrilateral Element, and 9-Node Quadrilateral Element was presented. The shape functions and the element equations related to each element were presented through a detailed mathematical formulation. Additionally, the Jacobian matrix for the second order derivatives was simplified and used to derive each element’s strain-displacement matrix in bending. The elements were compared using carefully selected elastic and aero-elastic bench mark problems, regarding the number of elements needed to reach convergence, the resulting accuracy, and the needed computation time. The best suitable element for elastic free vibration analysis was found to be the Linear Quadrilateral Element with deformation-based shape functions, whereas the most suitable element for stress analysis was the 8-Node Quadrilateral Element, and the most suitable element for aero-elastic analysis was the 9-Node Quadrilateral Element. Although the linear triangular element was the last choice for modal and stress analyses, it establishes more accurate results in aero-elastic analyses, however, with much longer computation time. Additionally, the nine-node quadrilateral element was found to be the best choice for laminated composite plates analysis.

Aerodynamic Optimization Based on Continuous Adjoint Method for a Flexible Wing

Aerodynamic optimization based on continuous adjoint method for a flexible wing is developed using FORTRAN 90 in the present work. Aerostructural analysis is performed on the basis of high-fidelity models with Euler equations on the aerodynamic side and a linear quadrilateral shell element model on the structure side. This shell element can deal with both thin and thick shell problems with intersections, so this shell element is suitable for the wing structural model which consists of two spars, 20 ribs, and skin. The continuous adjoint formulations based on Euler equations and unstructured mesh are derived and used in the work. Sequential quadratic programming method is adopted to search for the optimal solution using the gradients from continuous adjoint method. The flow charts of rigid and flexible optimization are presented and compared. The objective is to minimize drag coefficient meanwhile maintaining lift coefficient for a rigid and flexible wing. A comparison between the results from aerostructural analysis of rigid optimization and flexible optimization is shown here to demonstrate that it is necessary to include the effect of aeroelasticity in the optimization design of a wing.

Stable linear traction‐based equilibrium elements for elastostatics: Direct access to linear statically admissible stresses and quadratic kinematically admissible displacements for dual analysis

SummaryThis paper establishes the basic framework for the traction‐based equilibrium finite element method (traction‐based EFEM). Stable linear traction‐based equilibrium elements are formulated using the macro‐element technique. An arbitrary internal macro‐point renders a linear triangular element stable, while a stable linear quadrilateral element requires the macro‐point to locate at the intersection of diagonals. Then, a Lagrangian formulation is utilized to minimize the complementary energy under equilibrium constraints, and consequently, tractions as well as additional Lagrange multipliers are obtained. Linear statically admissible (SA) stresses are thereafter acquired from tractions. As for Lagrange multipliers, they turn out to coincide well with rigid‐body displacements in each element after simple modifications. With rigid‐body displacements and linear tractions known, quadratic displacements and the kinematically admissible (KA) counterpart thereof by recovery are obtainable. The knowledge of both SA stresses and KA displacements renders dual analysis directly applicable. That is to say, the traction‐based EFEM is featured with direct access to strict upper and lower bounds of strain energy and other quantities of interest. Copyright © 2014 John Wiley & Sons, Ltd.

Least-squares finite element solution of Euler equations with H-type mesh refinement and coarsening on triangular elements

Purpose – The purpose of this paper is to demonstrate successful use of least-squares finite element method (LSFEM) with h-type mesh refinement and coarsening for the solution of two-dimensional, inviscid, compressible flows. Design/methodology/approach – Unsteady Euler equations are discretized on meshes of linear and quadratic triangular and quadrilateral elements using LSFEM. Backward Euler scheme is used for time discretization. For the refinement of linear triangular elements, a modified version of the simple bisection algorithm is used. Mesh coarsening is performed with the edge collapsing technique. Pressure gradient-based error estimation is used for refinement and coarsening decision. The developed solver is tested with flow over a circular bump, flow over a ramp and flow through a scramjet inlet problems. Findings – Pressure difference based error estimator, modified simple bisection method for mesh refinement and edge collapsing method for mesh coarsening are shown to work properly with the LSFEM formulation. With the proper use of mesh adaptation, time and effort necessary to prepare a good initial mesh reduces and mesh independency control of the final solution is automatically taken care of. Originality/value – LSFEM is used for the first time for the solution of inviscid compressible flows with h-type mesh refinement and coarsening on triangular elements. It is shown that, when coupled with mesh adaptation, inherent viscous dissipation of LSFEM technique is no longer an issue for accurate shock capturing without unphysical oscillations.