Subparametric Transformation Research Articles

An efficient finite element computation using subparametric transformation up to cubic-order for curved triangular elements

PurposeA finite element computational methodology on a curved boundary using an efficient subparametric point transformation is presented. The proposed collocation method uses one-side curved and two-side straight triangular elements to derive exact subparametric shape functions.Design/methodology/approachOur proposed method builds upon the domain discretization into linear, quadratic and cubic-order elements using subparametric spaces and such a discretization greatly reduces the computational complexity. A unique subparametric transformation for each triangle is derived from the unique parabolic arcs via a one-of-a-kind relationship between the nodal points.FindingsThe novel transformation derived in this paper is shown to increase the accuracy of the finite element approximation of the boundary value problem (BVP). Our overall strategy is shown to perform well for the BVP considered in this work. The accuracy of the finite element approximate solution increases with higher-order parabolic arcs.Originality/valueThe proposed collocation method uses one-side curved and two-side straight triangular elements to derive exact subparametric shape functions.

An effective higher order finite element computation method for analyzing the eddy current losses in induction motors using subparametric transformations

An enhanced sub-parametric finite element method is presented for analyzing the eddy current losses in induction motors using higher order sub-parametric transformations. The purpose of this study is to construct a very efficient, simple, and exact higher order sub-parametric approach employing a 2D automated mesh generator implemented in JuliaFEM. Alternating current motors are employed in a wide range of industrial and household applications on a frequent basis such as in fans, water heaters, ovens, pumps and many more. High efficiency of the motors is one of the most desirable parameters of these motors. Eddy current losses are predominant in these motors which are responsible for a decline in the efficiency. Consequently, the use of eddy current analysis is vital in evaluating the behaviour of electric devices under various loads. This paper proposes a novel finite element method approach to minimize the 2D eddy current phenomenon by taking the diffusion equation into account. Eddy currents are the primary source of torque in various types of equipment, such as induction motors. In other cases, it is important to analyse the eddy currents to estimate the losses, reactance, and resistance.

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.

Pressure and flow transition over NACA airfoil with thrust optimized parabolic arcs

This paper presents an accurate prediction of surface drag, pressure coefficient and flow transition around a symmetric airfoil design through the implementation of finite element discretization. Skin friction coefficient has been analyzed at different Reynolds number and at various angles of attack for NACA0012 airfoil design. The transition from laminar to turbulent flow significantly impacts the separation of the boundary layer and skin friction of the airfoil, ultimately affecting its aerodynamic characteristics. The pressure distribution around the surface of NACA0012 has also been computed at several Reynolds number and different attack angles using higher order triangular meshes. Numerical evaluation of these results when compared with the experimental results demonstrates superior accuracy with higher-order finite element mesh. The investigation carried out by higher-order element meshing approach is based on the subparametric transformation of parabolic arcs of triangular element curved at one side. Moreover, the high aspect ratio meshes obtained during discretization of airfoil design have been considered for the analysis. The investigation has yielded results that closely align with experimental data, demonstrating excellent performance. The current efficient analysis of pressure and flow transition is beneficial in interpreting aerodynamic performance and flow simulations for aerospace and computational fluid dynamics applications.

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.

A new finite element method using MATLAB for Poisson’s Equation in axisymmetric domain

This paper demonstrates finite element procedure for two-dimensional axisymmetric domains. For many engineering applications like structural engineering, aerospace engineering, geo-mechanics etc., the solution domain and boundary conditions are axisymmetric. Henceforth, we can illuminate just the axisymmetric part of the solution domain that gives the data of the entire domain. This paper demonstrates the effectiveness of using MATLAB programming demonstrated by Persson et.al (2004) as the initial mesh for discretization of axisymmetric domains for higher order meshing. Further, solving some class of partial differential equations using finite element method with nodal relation given by subparametric transformations Rathod et.al (2008). In this paper a cubic order curved triangular meshing for some of the domains like ellipse and circle are demonstrated. These in turn finds its applications in the fields like stress analysis in mechanical engineering, torsion twist (shear strength) analysis in civil engineering, evaluation of stress intensity factor for quarter elliptical crack in pressure vessels in equipment industry etc,. The output data from the meshing scheme like meshing of the domain, nodal position, element connectivity and boundary edges are been used in the finite element procedures. The efficiency of the method is achieved by p-refinement scheme i.e., fixing the number of elements and increasing the polynomial order.

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.

An efficient automated higher-order finite element computation technique using parabolic arcs for planar and multiply-connected energy problems

A two-dimensional efficient and most accurate subparametric higher-order finite element technique are offered in this paper for some energy problems. It is used for the computation of eigenvalues over planar and multiply connected curved domains. This technique uses a high-quality and higher-order automated mesh generator developed from curvedHOmesh2d.m. The proposed mesh generator utilizes up to sextic-order (28-noded) one-sided curved triangular finite elements along with parabolic arcs to most accurately match the curved boundaries. One of the complete developed MATLAB code using the higher-order curved meshing technique for a challenging multiply-connected domain is provided for the readers. This computational technique is most accurate owing to the fact that higher-order finite elements are employed. Its efficiency can be witnessed in the drastic decrement of the computational time which has been attained by the use of the subparametric transformations with parabolic arcs. The degree of the Jacobian is of lower-order for each higher-order element compared to the conventional higher-order finite element method. This approach uses an excellent discretization procedure, the best quadrature rule, and an outstanding subparametric finite element process. Thus, the proposed approach enhances the accuracy of the numerical solution of eigenvalues occurring in several electromagnetic applications due to minimal curvature loss. The mathematical explanation of this process with its implementation for the effective computation of eigenvalues is described here. Several electromagnetic problems are known to have spurious solutions in the multiply-connected domains by many of the available numerical methods. Effective numerical results are obtained for these problems as illustrated in the provided examples with the proposed approach. These problems are shown to recognize the legitimacy of the present formulation. For the illustrative cases from the proposed technique, the numerical outcomes and best-published outcomes or analytical predictions are in great accord.

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.

Automated Mesh Generation Using Curved Cubic Triangular Elements for a Circular Domain with a Finite Element Implementation

We propose a method for automated unstructured mesh generation using curved cubic triangular elements for a circular domain which can be efficiently used for finite element analysis in industrial engineering and applied sciences. This approach uses subparametric transformations with the parabolic arcs to obtain the nodal relations for the curved geometry under consideration. Persson and Strang developed a simple and well-known MATLAB mesh generator distmesh2d with linear triangular elements using signed distance function for the geometric description. The technique used here, to generate cubic (10-noded) triangular elements is based on distmesh2d. This approach can be easily adapted for any curved geometry. For illustration purpose, in this paper finite element method is applied to a Poisson equation over a circular domain. The efficiency of the proposed technique using curved cubic triangular elements is shown in the numerical results which are more accurate compared to linear and straight edged cubic ordered triangular elements for a fixed element size.

MATLAB 2D higher-order triangle mesh generator with finite element applications using subparametric transformations

This paper presents a novel automated higher-order (HO) unstructured triangular mesh generation of the two-dimensional domain. The proposed HO scheme uses the nodal relations obtained from subparametric transformations with parabolic arcs, especially for curved geometry. This approach is shown to drastically simplify the computational complexities involved in the HO finite element (HOFE) formulation of any partial differential equation (PDE). The prospective generalised MATLAB 2D mesh generation codes, HOmesh2d for the regular domain and CurvedHOmesh2d for a circular domain are based on the MATLAB mesh generator distmesh of Persson and Strang. As an input, the code takes a signed distance function of the domain geometry and the desired order for the triangular elements and as outputs, the code generates an HO triangular mesh with element connectivity, node coordinates, and boundary data (edges and nodes). The working principle of HOFE scheme, using subparametric transformations with the proposed HO automated mesh generator is explained. The simplicity, efficiency, and accuracy of the HOFE method, with the proposed HO automated mesh generator up to 28-noded triangular elements, are illustrated with elliptic PDE. The proposed techniques are applied to some electromagnetic problems. The use of higher order elements from the proposed mesh generator is shown to increase the accuracy and efficiency of the numerical results. Also, with the proposed HOFE scheme it is verified that HO elements significantly decrease the numbers of degrees of freedom, and elements required to achieve a specific level of accuracy compared to lower order elements. Numerical results show that the HO elements outperform the lower order elements in terms of efficiency and accuracy of the numerical results.

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.

Synthetic division based integration of rational functions of bivariate polynomial numerators with linear denominators over a unit triangle {0⩽ ξ, η⩽1, ξ+ η⩽1} in the local parametric space ( ξ, η)

The domain of real problems in mechanics often contains curved boundaries. Curved boundaries are often more accurately modelled by curved finite elements than by straight edged elements, as straight sides are perfectly satisfactory if the domain has a polygonal boundary. Because fewer curved elements are required, the effort needed to obtain a solution is usually reduced. If some parts of the boundary are curved, however, elements with at least one curved side are desirable. Our aim in this paper is to consider the triangular element with two straight sides and one curved side. This paper is concerned with explicit formulae for evaluating integrals of rational functions of bivariate polynomial numerators with linear denominators over a unit triangle {0⩽ξ, η⩽1, ξ+η⩽1} in the local parametric two dimensional space ( ξ, η). These integrals arise in finite element formulations of second order linear partial differential equations by use of triangular element with two straight sides and one curved side of quadratic variation which often require relatively large numerical effort to integrate. The curved elements considered here are the four node, six node and ten node triangular elements with one curved side of quadratic variation and the other two sides have straight edges under the isoparametric and subparametric transformations, respectively. We have shown that by use of a method similar to synthetic division, the rational integrals of nth order bivariate polynomial numerator with a linear denominator having ( n+1)( n+2)/2 integrals can be reduced to rational integrals of nth order polynomial numerator in one variate with the same linear denominator having ( n+1) integrals and a simple integral of bivariate polynomial expression containing n( n+1)/2 terms (which is free from denominator), and this amounts to a substantial reduction in the numerical effort for such integrals. The explicit analytical integration formulae (obtained) up to sextic polynomial numerator in bivariates ξ, η due to different element geometry are, for clarity and reference, summarised in tables. Finally three application examples are also considered. For the first example we have used integration formulae derived in all theorems of this paper and explained the detailed computational scheme. For the other two examples computational scheme follows in a similar way. It is observed in the solution of all problems that the displacements and torsional constants are satisfactory when components of all element matrices are calculated by analytical integration formulae derived in this paper. It is also observed that in the calculation of components of element matrices by using numerical integration formulae (e.g., 7-point and 13-point rule) much discrepancy occurs if the element geometry is coarse with a concave curve side. But it is found that the analytical integration technique is always consistent. Therefore the symbolic integration formulae presented in this paper are reliable and may lead to an easy and systematic incorporation of element matrices required in the finite element solution procedure.

Buckling analysis of arbitrarily shaped plates by spline finite strip method

The spline finite strip method is applied to the buckling analysis of arbitrarily shaped plates. The plate is first mapped into a rectangular domain in the natural coordinate plane by the subparametric transformation, and the mapped plate is discretised into a number of strips. The displacements of each strip are described by interpolation functions which are given as products of piecewise polynomials and B-3 spline functions. The eigenvalue matrix equation for the buckling analysis is then formulated and solved by the same procedure as that of the standard finite element method. Numerical examples are presented to demonstrate the versatility and accuracy of the method.

Higher order subparametric elements for large deflection analysis of plates

A family of higher-order subparamctric elements has been developed for large deflection analysis of flat plates of any geometric shapes subjected to lateral loading. The formulation procedure for these elements with 17–25 nodes involving in-plane as well as bending displacements with subparametric transformation is described and the derivation of the tangential stiffness matrix based on the classical von Karman nonlinear large deflection theory is presented. The Newton-Raphson scheme with modification is used. Several numerical examples of skewed slabs and circular plates are worked out and the results are compared with available data given by other research workers. It is found that these higher-order elements are simple, versatile, economical and convenient to use and give accurate results for large deflection analysis of plates.

A collocation finite element method for potential problems in irregular domains

AbstractA potentially powerful numerical method for solving certain boundary value problems is developed. The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements. Bicubic Hermite elements with four degrees‐of‐freedom per node are used. A subparametric transformation permits the precise positioning of the collocation points for maximum accuracy as well as a unique representation of irregular boundaries. It is shown that by taking advantage of the boundary conditions, a minimum number of collocation points can be used.The method is particularly suitable for potential and mass transport problems where a C1 continuous solution is required. In contrast to the Galerkin approach, it does not require the evaluation of basis function products and numerical integration, also the coefficient matrix contains only about half as many non‐zero terms as the corresponding Galerkin coefficient matrix. This results in approximately a 90 per cent reduction in formulation and a 50 per cent reduction in solution operation, as compared with the Galerkin finite element method, for this type of problem. Examples show that the accuracy of the collocation solution is as good as or better than that of the Galerkin solution.