Order Triangular Elements Research Articles

On an efficient octic order sub-parametric finite element method on curved domains

An efficient finite element method using the sub-parametric transformations of the curved boundaries for the 45-node octic order triangular element is presented in this paper. The one-sided curved and two-sided straight triangle in the universal coordinate system (x,y) is mapped onto an isosceles triangle (ξ,η)∈[0,1] and ξ+η≤1 in the (ξ,η)-coordinate system by using an isoparametric coordinate transformation. Using such a transformation, the curved edge of the triangular element is altered implicitly by a newly defined octic curve. Such an octic curve is obtained by using the topology of the curved edge. With an apparent condition of arc being a parabola, a new transformation is developed to find a unique curve for each edge that passes through the points on the original curved arc. Overall, the new sub-parametric transformations are developed, these include the transformation for the curved edge on the original boundary, in addition to the parameters from the curved triangle's interior. Several numerical experiments are presented to show the efficacy of the proposed method. The computational results obtained for a boundary value problem using octic order basis functions outperform the results obtained using the existing higher order basis functions such as quadratic (6-node), cubic (10-node), quartic (16-node), quintic (21-node), sextic (28-node), and septic (36-node) order triangular elements (TEs) with curves. The method implemented in this paper can be easily extendable to quasi-static and dynamic crack evolution problems within linear and nonlinear elasticity.

A consistent projection integration for Galerkin meshfree methods

An efficient and inherently consistent integration method, named projection integration(PI), is proposed for Galerkin meshfree methods with arbitrary order. In contrast to traditional numerical integration methods, PI defines a projection operator based on a weighted norm to systematically approximate meshfree shape functions in an appropriate and consistent function space. Then a substitute function space is established by utilizing projected shape functions as a basis, and Galerkin weak form is approximately calculated by substituting the projected shape functions for the original meshfree shape functions. Owing to the projection consistency of the projection operator, the projected shape functions can still meet consistent conditions. For convenient implementations, the triangular finite element space is selected to approximate meshfree shape functions and the weight function in the weighted norm is carefully chosen to simplify the projection operator in this paper. Arbitrary order integration constraint conditions can be easily met by utilizing the corresponding order triangular finite element shape functions with low order Gauss quadrature rules, which means the optimal convergence rate can be obtained. A series of numerical examples are presented to demonstrate the proposed method’s efficiency and superior convergence performance.

Evaluation of wing performance for NACA 4415 using subparametric finite element transforms

The goal of the current research is to improve the lift performance of an aerodynamic device for commercial application by performing accurate aerodynamic computations and flow analyses of NACA 4-digit airfoil designs. An enhanced aerodynamic characteristic, which has been estimated effectively by interpreting the impacts of Reynolds numbers at various angles of attack, is crucial for a cost-effective airfoil shape. Numerous Reynolds numbers have been used in the analysis of these aerodynamic properties. Investigations were carried out using subparametric finite elements and a unique approach for higher order triangular element meshing in relation to the NACA 4415 airfoil design. This approach provides precise boundary node information for the airfoil design, resulting in improved calculations of the lift coefficient, drag coefficient, and lift-to-drag ratio. Moreover, it effectively addresses the challenges faced in aerospace research pertaining to flow analysis, thereby reducing the complexities involved. The evaluationof the airfoil shape and analysis for incompressible flow over the NACA 4415 airfoil design could improve the overall effectiveness of the aerodynamic vehicle. Any vehicle's aerodynamic surface must go through a number of tests before being launched. Modern technology can be incorporated into the design process to reduce the high costs associated with these testing processes. The current flow analysis and aerodynamic investigations exhibit a high degree of accuracy, and the use of HO elements lowers the expense of the experiment.

Function and Curl Recovery for the Lowest Order Triangular Edge Element

Function and Curl Recovery for the Lowest Order Triangular Edge Element

Studying the Effects of Negative Skin Friction on Single Piles in Basrah Governorate

The finite element method capable of simulating the behavior of deep foundations subjected to negative skin friction in Basrah soil is investigated. Single piles under drag forces are analyzed using the PLAXIS program with an axisymmetric model. Linear elastic, Soft Soil and Mohr-Coulomb constitutive relations are adopted, where higher order triangular element is chosen for pile and soil clusters. Both pile and soil are modeled using (15)-node triangular elements. Three sites in Basrah province (Umm Qasr Port, Khor Al-Zubair, and Shatt AlArab Hotel) were selected to perform this study. The soil profile and layer characteristics are obtained from the soil investigation reports. Where the negative skin friction is evaluated due to filling loads. It is Conclusion thatSmall relative displacements are necessary to activate the negative skin friction. The elastic shorting for pile effect negative skin friction, due to increase relative displacement. The elastic shorting of the driven pile is more than that of the bored pile due to the less cross-sectional area of the driven pile. The results revealed proportional relation between the developed drag forces and pile section dimensions, interface friction factor, and fill height, with a maximum effect on the section dimension and minimum effect on the interface factor. The locations of neutral points are not sensitive to the above-mentioned factors.

L2-Norm Based a Posteriori Error Estimates of Compressible and Nearly-Incompressible Elastic Finite Element Solutions

The displacement and stress-based error estimates in a posteriori error recovery of compressible and nearly-incompressible elastic finite element solutions is investigated in the present study. The errors in the finite element solutions, i.e., in displacement and stress, at local and global levels are computed in L2-norm of quantity of interest, namely, displacements and gradients. The error estimation techniques are based on the least square fitting of higher order polynomials to stress and displacement in a patch comprising of node/elements surrounding and including the node/elements under consideration. The benchmark examples of compressible and incompressible elastic bodies, with known solutions employing triangular discretization schemes, are implemented to measure the finite element errors in displacements and gradients. The mixed formulation involving displacement and pressure is used for incompressible elastic analysis. The performance of error estimation is measured in terms of convergence properties, effectivity and mesh required for predefined precision. The error convergence rate, in FEM original solution, recovered solution using displacement recovery-based and stress-based error recovery technique for stresses, are obtained as (1.9714, 2.8999, and 2.5018) and (0.9818, 1.7805, and 1.4952) respectively for compressible and incompressible self-loaded elastic plate benchmark example using higher order triangular elements. It is concluded from the study that displacement fitting technique for extracting higher order derivatives shows a very effective technique for recovery of compressible and nearly-incompressible finite element analysis errors.

Investigation of aerodynamic characteristics of NACA0012 airfoil design using parabolic arcs

An investigation of the pitching moment coefficient has been performed numerically for NACA0012 airfoil design in this paper. The evaluation of the aerodynamic parameter has been analyzed with various ranges of Reynolds number from lower to higher (104 < Re <105). An essential aerodynamic parameter, the pitching moment coefficient is associated with torque developed by the aerodynamic forces around the airfoil design. A novel method of higher order triangular element meshing around the airfoil design has been utilized for the present analysis. The higher order triangular meshing is efficiently performed around the airfoil by incorporating the parabolic arcs of a one-sided curved triangular element. The curved triangular element satisfies the sub-parametric transformation of finite elements. Three different triangular elements such as linear, quadratic, and cubic elements have been considered to mesh around the NACA0012 airfoil design by MATLAB code. The higher order element has proved its efficiency in analyzing the parameter with less computational time and effort which is essential to enhance the performance of aerodynamic structures.

An automated higher order meshing for NACA0018 airfoil design using subparametric transformation

A novel higher order meshing around the airfoil design by using finite simple triangle elements has been presented in this paper. An automated meshing has been performed by linear to sextic order (n = 1, 2, 3, 4, 5 and 6) simple triangle elements using MATLAB code. The subparametric transformation approach has been implemented in the present accurate and efficient meshing. Linear, quadratic, cubic, quartic, quintic and sextic order triangle shaped elements are been utilized to discretize the area near the airfoil shape. An itemized extraction of the edge information, nodal coordinates, the total number of discretized elements and the triangle indices has been carried out from the present meshing. A NACA0018 airfoil design of 18% thickness of max camber is under consideration for the present work. Meshing is executed with the MATLAB code AirfoilHOmesh2d .m by implementing simple changes based on the present 4-digit NACA0018 airfoil profile. The accuracy of an eminent linear mesh generator scheme presented by Gilbert Strang and Persson has been proved and verified. Present work has extended these eminent linear meshing scheme by utilizing subparametric transformation on higher order triangular elements. The algorithm of meshing with higher order has been explained in detail and represented in flowchart form in the Appendix. The data extracted from the meshing can offer a simplified approach to optimize the airfoil shape, for the study of flow around the airfoil in fluid dynamics and the analysis of adverse effects on its external surface in aerospace applications. The advanced higher order meshing is beneficial for the optimization of any symmetric airfoil resulting a better economical fuel usage in the aircraft.

2D Higher order triangular mesh generation in irregular domain for finite element analysis using MATLAB

This paper presents an automated mesh generation for straight and curved sided irregular domains with unstructured two dimensional higher order triangular elements. The present higher order (HO) scheme has been implemented on the basis of subparametric transformations which are extracted from the nodal relations of parabolic arcs especially used for the curved domains. This new restructured meshing scheme is based on distmesh2d introduced by Persson and Gilbert Strang. In this work a higher order triangular mesh for two irregular domains star shaped domain and a circle inscribed in a rectangle has been constructed. These in turn is able to find its application in abundant flow problems and thermodynamics. Present innovative meshing scheme provides a refined and improved high quality meshes for these domains and produce accurate results of the node position, boundary edges and element connectivity for the discretized element. This is an advantage in executing finite element method with less computational efforts in practical engineering applications over the irregular domains.

Accurate higher order automated unstructured triangular meshes for airfoil designs in aerospace applications using parabolic arcs

This paper presents automatically generated higher order curved triangular meshes around airfoil design using MATLAB code. This work shows a valuable basis for the finite element procedures involved in evaluating aerodynamic performances. Finite element method (FEM) effectively solves all computational fluid dynamics problems around the airfoil and for that region around the airfoil that has been discretized with unstructured curved triangular elements. Meshes have been formed on the basis of subparametric transformation created for the curved triangular element obtained from the nodal relations of parabolic arcs. This scheme can be used to obtain the output data of node coordinates, element connectivity and boundary values for all discretized elements over the airfoil design. A spectacular work done on linear triangular element meshing over a domain by Persson and Gilbert Strang is the basis of present meshing scheme. The proposed meshing scheme presents a refined higher order (HO) curved triangular discretization of few airfoil designs namely NACA0012, NACA0015 and NACA0021 inscribed inside a circle. The approach of the suggested meshing scheme described in this paper can be applied to numerous aerospace applications such as computing pressure gradients, understanding atmospheric nature study, evaluating laminar viscous compressible flow around the airfoil shape, etc. The element and nodal information gained from this discretization is useful for the numerical solutions of FEM and for the aerodynamic portrayal. This paper is aimed at the innovative discretization scheme which can be extended to all kinds of NACA airfoil designs. We have provided the MATLAB code AirfoilHOmesh2d for HO curved meshing around an airfoil with a cubic order triangular element. The mathematical explanation of this along with the description and implementation of it on few airfoil designs is described. The flowchart of the MATLAB code for cubic order meshing over airfoil design has been provided. This implementation supports many applications in an aerodynamic performance that have been elaborated in this paper. Two applications for the analysis of potential flow around airfoil and computation of pressure coefficient (Cp) on the surface of the airfoil design have been performed. It has been verified and found that the present HO curved meshing technique efficiently gives converging solution.

Cubature versus Fekete–Gauss nodes for spectral element methods on simplicial meshes

In a recent JCP paper (by Y. Liu et al., vol. 336, p. 458, 2017) a higher order triangular spectral element method (T SEM) is proposed to address seismic wave field modeling. The main interest of this T SEM is that the mass matrix is diagonal, so that an explicit time marching becomes very cheap. This property results from the fact that, similarly to the usual SEM, the basis functions are Lagrange polynomials based on a set of points that shows both nice interpolation and quadrature properties. In the quadrangle, i.e. for the QSEM, the set of points is simply obtained by tensorial product of Gauss-Lobatto-Legendre (GLL) points. In the triangle, finding such an appropriate set of points is however not trivial. Thus, the recent JCP work follows anterior works that started in 2000's and now provides cubature nodes and weights up to N = 9, where N is the total degree of the polynomial approximation. Here we wish to evaluate the accuracy of this cubature nodes T SEM with respect to the Fekete-Gauss one, that makes use of two sets of points, namely the Fekete points and the Gauss points of the triangle for interpolation and quadrature, respectively. Because the Fekete-Gauss T SEM is in the spirit of any nodal hp-finite element methods, one may expect that the conclusions of this Note will remain relevant if using other sets of carefully defined interpolation points.

Vibration of thermally post-buckled hybrid laminates with two non-uniformly distributed fibers

This study presents the first vibration analysis of thermally post-buckled hybrid laminates with non-uniformly distributed graphite and E-glass fibers in a single lamina. A 54 degree-of-freedom high order triangular plate element is developed based on the Von Karman large deflection assumption. The formulation of the location dependent linear, nonlinear stiffness and mass matrices due to non-homogeneous material properties, geometrical stiffness matrix due to thermal effects and thermal moment vector due to temperature gradient is derived. The effects of hybrid-fiber distribution on the natural frequencies and mode shapes of thermally post-buckled laminates are discussed. The numerical results reveal that the redistribution of two fibers can considerably decrease the postbuckling deflections of the hybrid laminates and significantly modify the natural frequencies. The stiffening effect of fiber redistribution is more obvious for the laminate with a higher volume fraction index and clamped edges. Special buckling and vibration mode shapes are observed, along with multiple vibration mode shifting.

Stress Analysis of Free-Edge Laminated Composite Plates by Two Bending Elements

Two higher order triangular elements will be formulated for inter-laminar stresses analysis in free edge of the laminated composite bending plates. The analysis is based on a displacement model and includes the higher order shear deformation. To demonstrate the validity and efficiency of the proposed method, the orthotropic laminated composite plates with arbitrary laminations and boundary conditions will be analyzed. In this paper, the effects of increasing the number of nodes and the degree of interpolation functions will be studied. It will be proven that by using particular higher order interpolation, more accurate stress resultants for plates can be predicted. Furthermore, a good rate of convergence will be found by utilizing only a very coarse mesh.

Finite Element Solution of Darcy–Brinkman Equation for Irregular Cross-Section Flow Channel Using Curved Triangular Elements

The finite element method of solution with optimal subparametric higher-order curved triangular elements is used to solve the 3-D fully developed Darcy–Brinkman flow equation through channel of irregular cross-section. Extensive numerical computation and numerical experimentation are done using the quadratic, cubic, quartic and quintic order triangular elements, which reveals that the parameters’ influence on the velocity distributions are qualitatively similar for all the cross-sections irrespective of whether they are of regular or irregular cross-sections. The quintic order curved triangular element yields the solution of a desired accuracy of 10-6. The method can be easily employed in any other irregular cross-section channels.

Aerothermoelastic analysis of composite laminates with variable fiber spacing

This study presents the effects of variable fiber spacing on the thermal postbuckling, vibration, and flutter behaviors of composite laminates subjected to aerodynamic force and thermal stress. A 54 degree-of-freedom high order triangular plate element is developed based on the Von Karman large deflection assumptions and quasi-steady supersonic aerodynamic theory. The numerical results reveal that the redistribution of fibers can dramatically increase the critical buckling temperature, and efficiently increase the natural frequencies and flutter boundary. The sequence of the natural mode may be altered and the frequency coalescence pair may change. The stability boundary is different and some specific phenomena are discussed.

Seleção de topologias ótimas de estruturas elásticas 2D com restrição de tensão – via Smooth Evolutionary Structural Optimization

This paper approaches the topology optimization problems in plane linear elasticity considering the minimization of the volume with restriction of the stress employing an index of performance for monitoring the meeting of the optimum region. It is used for this purpose the classical evolutionary structural optimization, or ESO ‐ evolutionary structural optimization. This procedure is based on systematic and gradual removal of the elements with lower stress compared with the maximum stress of the structure. This procedure also known as a process “hard‐kill”. It is proposed a variant of the ESO method, called SESO ‐ Smoothing ESO, which is based on the philosophy that if an element is not really necessary for the structure, its contribution to the structural stiffness will gradually diminish until it has no longer influence in the structure, so its removal is performed smoothly. That is, their removal is done smoothly, reducing the values of the constitutive matrix of the element as if it were in the process of damage. A new performance index for the monitoring of this evolutionary process smoothed is proposed herein. The applications of ESO and SESO are made with the finite element method, but considering a high order triangular element based on the free formulation. Finally, it is implemented a spatial filter in terms of stress control, which was associated with SESO technique proved to be very stable and efficient in eliminating the formation of the checkerboard.

Parallelization of Enriched Free Mesh Method for Large Scale Fluid-structure Coupled Analysis

This paper proposes a new treatment for parallel EFMM. EFMM has a shortcoming that is too difficult to apply into adaptive problem or dynamic problem. In the fluid-structure coupled analysis field, these methods are using as a structure analysis method. The new fluid-structure coupled analysis method to solve large-scale fluid-structure interaction problem is already proposed by us. Our new method has nodal consistency at the fluid-structure interface and its calculation efficiency and accuracy are high. SUPG/PSPG stabilized FEM is adopted for fluid analysis, while the high-accuracy analysis method based on EFMM developed by the authors is adopted for structure analysis. EFMM has a feature that can be possible to obtain high accuracy analysis result without high order elements. In short, as the common feature of these methods, it is possible to analyze a fluid or a structure rather accurately by using the first order triangular or tetrahedral elements. In addition, variables are exchanged exactly at the common nodes on the fluid-structure boundary without deteriorating accuracy and calculation efficiency due to the interpolation of variables between nodes. From these features, SUPG/PSPG stabilized FEM and EFMM have a very good chemistry. Therefore, our proposed method can be possible to obtain high accuracy fluid-structure coupled analysis result without high order element. However, our proposed method had a shortcoming that is appeared when it is introduced into parallel method. This shortcoming occurs due to the characteristics of the EFMM. Specifically, parallel efficiency that is very important factor of large-scale analysis will be decrease. Generally, domain decomposition method is certainly needed in the large-scale analysis field. But, usual domain decomposition method will not be able to introduce. Because, making stiffness matrix process by EFMM is different from conventional FEM. Therefore, node data for making stiffness matrix of local element cluster is across some domains. To obtain these data, unnecessary communications will be occurred. By this shortcoming, compatibility of SUPG/PSPG stabilized FEM and EFMM becomes very bad. In view of the large-scale analysis by our proposed method in the near future, this is a very important issue. In order to introduce our proposed method into large-scale parallel analysis, new treatment for parallel EFMM is necessary. Therefore, we propose the new treatment for parallel EFMM and verification of our new treatment and numerical example of fluid-structure interaction computed by our new method are described in this paper.

Modeling of Losses Due to Inter-Laminar Short-Circuit Currents in Lamination Stacks

Abstract The cores of electrical machines are generally punched and laminated to reduce the eddy current losses. These manufacturing processes such as punching and cutting deform the electrical sheets and deteriorate its magnetic properties. Burrs are formed due to plastic deformation of electrical sheets. Burr formed due to punching on the edges of laminated sheets impairs the insulation of adjacent sheet and make random galvanic contacts during the pressing of stacked sheets. The effect of circulating current occurs if the burrs occur on the opposite edges of the stacks of laminated sheets and incase of bolted or wielded sheets, induced current return through it. This induced current causes the additional losses in electrical machine. The existence of surface current on the boundary between two insulated regions causes discontinuity of tangential component of magnetic field. Hence, based on this principle, the boundary layer model was developed to study the additional losses due to galvanic contacts formed by burred edges. The boundary layer model was then coupled with 2-D finite element vector potential formulation and compared with fine mesh layer model. Fine mesh layer model consists of finely space discretized 950028 second order triangular elements. The losses were computed from two models and were obtained similar at 50 Hz. The developed boundary layer model can be further used in electrical machines to study additional losses due to galvanic contacts at the edges of stator cores.

Efficient finite element analysis of models comprised of higher order triangular elements

This paper introduces an efficient method for the finite element analysis of models comprised of higher order triangular elements. The presented method is based on the force method and benefits graph theoretical transformations. For this purpose, minimal subgraphs of predefined special patterns are selected. Self-equilibrating systems are then constructed on these subgraphs leading to sparse and banded null basis. Finally, well-structured flexibility matrices are formed for efficient finite element analysis.

Advantages of cubic arcs for approximating curved boundaries by subparametric transformations for some higher order triangular elements

In the finite element method, the most popular technique for dealing with curved boundaries is that of isoparametric coordinate transformations. In this paper, the 10-node (cubic), 15-node (quartic) and 21-node (quintic) curved boundary triangular elements having one curved side and two straight sides are analyzed using the isoparametric coordinate transformations. By this method, these curved triangles in the global coordinate system are mapped into a isosceles right angled unit triangle in the local coordinate system and the curved boundary of these triangular elements are implicitly replaced by cubic, quartic, and quintic arcs. The equations of these arcs involve parameters, which are the coordinates of points on the curved side. Relations are deduced for choosing the parameters in quartic and quintic arcs in such a way that each arc is always a cubic arc which passes through four points of the original curve, thus ensuring a good approximation. The point transformations thus determined with the above choice of parameters on the curved boundary and also in turn the other parameters in the interior of curved triangles will serve as a powerful subparametric coordinate transformation for higher order curved triangular elements. Numerical examples are given to demonstrate the accuracy and efficiency of the method.