Qualification: 
Ph.D, MPhil, MSc
Email: 
kv_nagaraja@blr.amrita.edu

Dr. K. V. Nagaraja currently serves as Professor at the Department of Mathematics, School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru campus. He has been actively guiding Ph.D. scholars in interdisciplinary areas and has authored around 50 technical papers published in National and International journals and also in conferences. His research interests are Finite Element Analysis, Computational Electromagnetics, Aerospace Applications, Biomechanics and Neural Networks. He is also a member of the editorial board for many journals.

Education

  • 2005: Ph. D.
    Bangalore University
  • 1998: M. Phil.
    Bangalore University
  • 1996: M. Sc.
    Bangalore University

Professional Appointments

Year  Affiliation
2011 - Present Professor, Department of Mathematics, School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru
2007-2011 Associate Professor & HoD, Department of Mathematics, School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru.
2005-2007 Assistant Professor  & HoD, Department of Mathematics, School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru.
2004-2005 Sr. Lecturer, Department of Mathematics, School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru.
2002-2004 Lecturer, Department of Mathematics, School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru.
1997-2002 Lecturer, Department of Mathematics, Golden Valley Institute of Technology, Kolar Gold Fields.

Research Interests

  • Numerical Analysis, Finite Element Method, Computational Electromagnetics, Computational Biology, Aerospace Engineering, Computational Fluid Dynamics, Artificial Neural Networks

Membership in Professional Bodies

  • Life Member of the Indian Society of Theoretical & Applied Mechanics (ISTAM), IIT Kharagpur

Publications

Publication Type: Journal Article

Year of Publication Title

2021

T. V. Smitha and Dr. K.V. Nagaraja, “MATLAB automated higher-order tetrahedral mesh generator for CAD geometries and a finite element application with the subparametric mappings”, Materials Today: Proceedings, vol. 42, pp. 330-342, 2021.[Abstract]


In this article MATLAB code is proposed for three-dimensional CAD geometries to produce automatic higher-order (HO) tetrahedral mesh. The HO mesh generator HOtetramesh proposed here is based up on a MATLAB PDE toolbox mesh generator function generateMesh. The input for this code is identical with that for input for the PDE toolbox mesh generator and the tetrahedral element with the necessary order. An HO tetrahedral mesh is generated as the output with the node coordinates, connectivity matrix, boundary edges and boundary nodes. The proposed mesh generator could be used efficiently to produce high-quality meshes in 3D finite element applications. A numerical example is provided with the complete detail for the demonstration of the proposed automated HO tetrahedral mesh generator over a 3D geometry by the subparametric finite element approach. This methodology can be used easily and reliably to address several problems in science and engineering.

More »»

2021

T. V. Smitha and Dr. K.V. Nagaraja, “Automated 3D higher-order tetrahedral mesh generator utilizing the subparametric transformations in MATLAB”, AIP Conference Proceedings, vol. 2316, p. 040003, 2021.[Abstract]


In this work, a novel automated higher-order (HO) unstructured tetrahedral mesh generators for three dimensional geometries are proposed. The proposed mesh generators, HOmesh3d for the regular geometries and CurvedHOmesh3d for spherical geometries are based on the very powerful mesh generator distmeshnd in MATLAB, developed by Persson. The developed MATLAB 3D mesh generation code CurvedHOmesh3d focus on the curved surface using the relations of the nodes acquired from the subparametric mapping by parabolic arcs. The input requirement for these codes is the same as that of distmeshnd and the required order of the tetrahedral elements. As output, an HO tetrahedral mesh with coordinates of each node, element connectivity matrix, boundary edges, and boundary nodes are generated. The effectiveness of CurvedHOmesh3d is enhanced by utilizing HO curved tetrahedral elements along the curved surface and regular HO tetrahedral elements within the interior of the problem domain. For curved geometries, this meshing approach can be very effectively employed with curved finite elements. The suggested mesh generator could be efficiently used for 3D finite element applications as it produces high-quality meshes with minimal curvature loss. This mesh could therefore be of very effective use for the solution of partial differential equations emerging in mechanical and aerospace engineering using the finite element method.

More »»

2020

S. Devi and Dr. K.V. Nagaraja, “An automated higher order meshing for NACA0018 airfoil design using subparametric transformation”, Materials Today: Proceedings, 2020.[Abstract]


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.

More »»

2020

P. G. Kannan and Dr. K.V. Nagaraja, “An Efficient Automatic Mesh Generator With Parabolic Arcs in Julia for Computation of TE and TM Modes for Waveguides”, IEEE Access, vol. 8, pp. 109508-109521, 2020.[Abstract]


An enriched finite element method is presented to numerically solve the eigenvalue problem on electromagnetic waveguides governed by the Helmholtz equation. In this work, a highly efficient, simple and precise higher order subparametric method was developed using a 2D automated mesh generator performed with JuliaFEM. The transcendence computerized discretization code in Julia is developed for the present work. For curved waveguide structures, meshes with one side and curving higher orders are proposed with triangular elements with parabolic arcs. The technique is shown for distinct waveguide constructions, and the results are compared with the strongest numerical or analytical results available. The results demonstrate that the proposed methodology is effective and accurate for generating finite element simulations for complex structures with black holes and irregular topology due to no curvature loss. This article presents a finding cutoff frequency performed with JuliaFEM-an open-source program. Analysis results produced by commercial software are considered for the comparison and show that the calculation results between the two programs do not differ significantly. This procedure can be used to achieve the most effective transmission of energy for electromagnetic applications.

More »»

2019

T. V. Smitha and Dr. K.V. Nagaraja, “An efficient automated higher-order finite element computation technique using parabolic arcs for planar and multiply-connected energy problems”, Energy, vol. 183, pp. 996-1011, 2019.[Abstract]


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. © 2019 Elsevier Ltd

More »»

2019

T. V. Smitha and Dr. K.V. Nagaraja, “Application of automated cubic-order mesh generation for efficient energy transfer using parabolic arcs for microwave problems”, Energy, vol. 168, pp. 1104-1118, 2019.[Abstract]


This paper aims to offer an efficient, simple and accurate cubic-order subparametric finite element technique utilizing 2-D automated mesh generator for microwave applications. The proposed technique utilizes the best discretization procedure, the finest quadrature rule and an excellent subparametric finite element algorithm for obtaining the numerical solutions of the Helmholtz equation. The high-quality automated mesh generator MATLAB code developed for the present work using HOmesh2d.m and CurvedHOmesh2d.m are provided. This approach utilizes up to cubic-order triangular mesh for arbitrary waveguide structures. The meshes with one side curved cubic-order triangular elements are proposed with parabolic arcs for curved waveguide structures. For regular waveguide structures with sharp edges having singularities, the utilization of unstructured cubic-order triangular meshes with refinement for the technique is proposed. The method is demonstrated for six different waveguide structures and the outcomes obtained are compared with the best available numerical or analytical results. The results show that the proposed technique produces an efficient and most accurate finite element results for the finite element analysis performed for waveguide structures with singularities and curved geometries as there is no curvature loss. Thus, the proposed subparametric finite element technique with the automated mesh generator is verified to yield a viable approach for getting the most precise numerical result for the 2-D Helmholtz equation in distinct waveguide cross-sections. Therefore, this approach can be used to produce the most efficient energy transfer for microwave applications.

More »»

2017

Sarada Jayan and Dr. K.V. Nagaraja, “Generalized Gaussian quadrature rules over an n-dimensional ball”, Pakistan Journal of Biotechnology, vol. 14, no. 3, pp. 423-428, 2017.

2016

Sarada Jayan and Dr. K.V. Nagaraja, “An optimal numerical integration method over a lune by using an efficient transformation technique”, Proceedings of the Jangjeon Mathematical Society, vol. 19, pp. 486-492, 2016.[Abstract]


In this paper, we derive an optimal numerical integration method to integrate functions over a lunar model, a closed region bounded by two different circular boundaries. The region is discretized into two and suitable efficient transformations are used to transform the regions to a zero-one square. After the transformation, a product formula is applied to derive the proposed numerical integration method. The generalized Gaussian quadrature nodes and weights for one dimension are used in the derived integration formula for evaluating the results. The results obtained for seven different functions are tabulated along with a comparative study in order to show that the proposed method gives more accurate results using less number of quadrature points and is the optimal one. More »»

2016

Dr. K.V. Nagaraja, Panda, T. Darshi, and Dr. V. Kesavulu Naidu, “Optimal subparametric finite elements for the computation of cutoff wavenumbers in waveguides”, AIP Conference Proceedings, vol. 1715, p. 020048, 2016.[Abstract]


In the present work, the computation of cutoff wavenumbers in waveguides with both straight and curved edge boundaries have been carried out using subparametric transformations. As compared to the conventional finite element methods, the subparametric transformation takes the advantage of mapping curved boundaries with greater accuracy. Under this transformation, any triangle with two straight sides and one curved side can be mapped to a standard right-angled triangle. This method has been applied to a regular L-shaped rectangular waveguide and also on an irregular curved geometry. The obtained cutoff frequencies of regular geometry are in close agreement with the existing values found in literature and those of irregular boundary have converged very well.

More »»

2015

T. Darshi Panda and Dr. K.V. Nagaraja, “Finite Element Method for Solving Eigenvalue Problem in Waveguide Modes”, Global Journal of Pure and Applied Mathematcs, vol. 11, no. 3, pp. 1241–1251, 2015.

2015

Sarada Jayan and Dr. K.V. Nagaraja, “Numerical integration over irregular domains using generalized Gaussian quadrature”, Proceedings of the Jangjeon Mathematical Society, vol. 18, pp. 21–30, 2015.

2015

Sarada Jayan and Dr. K.V. Nagaraja, “Numerical Integration over Three-Dimensional Regions Bounded by One or More Circular Edges”, Procedia Engineering, vol. 127, pp. 347–353, 2015.[Abstract]


A new integration method is proposed for integration of arbitrary functions over regions having circular boundaries. The method is developed using a new non-linear transformation which can transform such a region to a zero-one cube. The derivation of this formula over a circular and elliptic cylinder, cone and paraboloid is shown with numerical results.

More »»

2015

Dr. V. Kesavulu Naidu, Siddheshwar, P. G., and Dr. K.V. Nagaraja, “Finite Element Solution of Darcy–Brinkman Equation for Irregular Cross-Section Flow Channel Using Curved Triangular Elements”, Procedia Engineering, vol. 127, pp. 301–308, 2015.[Abstract]


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.

More »»

2015

Sarada Jayan and Dr. K.V. Nagaraja, “A General and Effective Numerical Integration Method to Evaluate Triple Integrals Using Generalized Gaussian Quadrature”, Procedia Engineering, vol. 127, pp. 1041–1047, 2015.[Abstract]


A general and effective numerical integration formula to evaluate all triple integrals with finite limits is proposed in this paper. The formula is derived by transforming the domain of integration to a zero-one cube. The general derivation along with results over specific regions like cuboid, tetrahedron, prism, pyramid and few regions having planar and non-planar faces is provided. Numerical results also are tabulated to validate the formula.

More »»

2014

Dr. K.V. Nagaraja, Dr. V. Kesavulu Naidu, and Siddheshwar, P. G., “Optimal Subparametric Finite Elements for Elliptic Partial Differential Equations Using Higher-Order Curved Triangular Elements”, International Journal for Computational Methods in Engineering Science and Mechanics, vol. 15, pp. 83-100, 2014.[Abstract]


This paper presents the finite element method using parabolic arcs for solving elliptic partial differential equations (PDEs) over regular and irregular geometry, which has many applications in science and engineering. Some numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. The results obtained are in excellent agreement with the exact values.

More »»

2014

Sarada Jayan and Dr. K.V. Nagaraja, “Numerical integration over n-dimensional cubes using generalized gaussian quadrature”, Proceedings of the Jangjeon Mathematical Society, vol. 17, pp. 63-69, 2014.[Abstract]


This paper gives a numerical integration rule for integrating functions over any n-dimensional cube. The rule is derived using a simple linear transformation of the given n-cube to a zero-one cube. The prescribed method is proved to be superior in a certain sense to the existing integration formulae. The performance of the method is illustrated for different type of integrands over different n-dimensional cubes.

More »»

2013

Dr. V. Kesavulu Naidu and Dr. K.V. Nagaraja, “Advantages of cubic arcs for approximating curved boundaries by subparametric transformations for some higher order triangular elements”, Applied Mathematics and Computation, vol. 219, pp. 6893-6910, 2013.[Abstract]


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. © 2013 Elsevier Inc. All rights reserved.

More »»

2012

Dr. K.V. Nagaraja and Sarada Jayan, “Generalized Gaussian quadrature rules over regions with parabolic edges”, International Journal of Computer Mathematics, vol. 89, pp. 1631-1640, 2012.[Abstract]


<p>This paper presents a generalized Gaussian quadrature method for numerical integration over regions with parabolic edges. Any region represented by R 1={(x, y)| a≤x≤b, f(x) ≤y≤g(x)} or R2={(x, y)| a≤y≤b, f(y) ≤x≤g(y)}, where f(x), g(x), f(y) and g(y) are quadratic functions, is a region bounded by two parabolic arcs or a triangular or a rectangular region with two parabolic edges. Using transformation of variables, a general formula for integration over the above-mentioned regions is provided. A numerical method is also illustrated to show how to apply this formula for other regions with more number of linear and parabolic sides. The method can be used to integrate a wide class of functions including smooth functions and functions with end-point singularities, over any two-dimensional region, bounded by linear and parabolic edges. Finally, the computational efficiency of the derived formulae is demonstrated through several numerical examples. © 2012 Copyright Taylor and Francis Group, LLC.</p>

More »»

2011

H. .T.Rathod, Dr. K.V. Nagaraja, Nagabhusan, C. S., and .Y.Shrivalli, H., “Symbolic Computation of high order Gauss Lobatto Quadrature formulas with variable precision”, International e-journal of Numerical Analysis and Related topics, Vol. 6, March 2011, PP. 52-, vol. 6, pp. 52–85, 2011.

2011

Sarada Jayan and Dr. K.V. Nagaraja, “Generalized Gaussian quadrature rules over two-dimensional regions with linear sides”, Applied Mathematics and Computation, vol. 217, pp. 5612-5621, 2011.[Abstract]


{This paper presents a generalized Gaussian quadrature method for numerical integration over triangular, parallelogram and quadrilateral elements with linear sides. In order to derive the quadrature rule, a general transformation of the regions

More »»

2011

H. Ta Rathod, Dr. B. Venkatesh, and Dr. K.V. Nagaraja, “Gauss Legendre - Gauss Jacobi quadrature rules over a Tetrahedral region”, International Journal of Mathematical Analysis, vol. 5, pp. 189-198, 2011.[Abstract]


This paper presents a Gaussian quadrature method for the evaluation of the triple integral I = ∫∫∫/T f (x, y, z) dxdydz, where f (x, y, z) is an analytic function in x, y, z and T refers to the standard tetrahedral region:{(x, y, z) | 0 ≤ x, y, z ≤1, x + y + z ≤1} in three space (x, y, z). Mathematical transformation from (x, y, z) space to (u, v, w) space maps the standard tetrahedron T in (x, y, z) space to a standard 1-cube: {(u,v,w) / 0 ≤ u, v, w ≤1} in (u, v, w) space. Then we use the product of Gauss-Legendre and Gauss-Jacobi weight coefficients and abscissas to arrive at an efficient quadrature rule over the standard tetrahedral region T. We have then demonstrated the application of the derived quadrature rules by considering the evaluation of some typical triple integrals over the region T.

More »»

2010

H. T. Rathod, Shrivalli, H. Y., Dr. K.V. Nagaraja, and Dr. V. Kesavulu Naidu, “On a New Cubic Spline Interpolation with Application to Quadrature”, Int. Journal of Math. Analysis, vol. 4, no. 28, pp. 1387–1415, 2010.[Abstract]


This paper presents a formulation and a study of an interpolatory cubic spline which is new and akin to the Subbotin quadratic spline. This new cubic spline interpolates at the first and last knots and at the two points located at trisections between the knots. Application of the proposed spline to integral function approximations and quadrature over curved domains are investigated. Numerical illustrations, sample outputs and MATLAB programs are appended.. More »»

2010

H. .T.Rathod, Gali, A., Shivaram, K. T., and Dr. K.V. Nagaraja, “Some composite numerical integration schmes for an arbitrary linear convex quadrilateral region”, International e-journal of Numerical Analysis and Related topics, vol. 4, pp. 19–58, 2010.

2010

Dr. K.V. Nagaraja and Rathod, H. Tb, “Symmetric Gauss Legendre quadrature rules for numerical integration over an arbitrary linear tetrahedra in Euclidean three-dimensional space”, International Journal of Mathematical Analysis, vol. 4, pp. 921-928, 2010.[Abstract]


In this paper it is proposed to compute the volume integral of certain functions whose antiderivates with respect to one of the variates (say either x or y or z) is available. Then by use of the well known Gauss Divergence theorem, it can be shown that the volume integral of such a function is expressible as sum of four integrals over the unit triangle. The present method can also evaluate the triple integrals of trivariate polynomials over an arbitrary tetrahedron as a special case. It is also demonstrated that certain integrals which are nonpolynomial functions of trivariates x, y, z can be computed by the proposed method. Then we have applied the symmetric Gauss Legendre quadrature rules to evaluate the typical integrals governed by the proposed method.

More »»

2010

Dr. K.V. Nagaraja, Dr. V. Kesavulu Naidu, and Rathod, H. Tb, “The use of parabolic arc in matching curved boundary by point transformations for sextic order triangular element”, International Journal of Mathematical Analysis, vol. 4, pp. 357-374, 2010.[Abstract]


This paper is concerned with curved boundary triangular element having one curved side and two straight sides. The curved element considered here is the 28-node (sextic) triangular element. On using the isoparametric coordinate transformation, the curved triangle in the global (x, y) coordinate system is mapped into a standard triangle: {(ξ,η) / 0 ≤ ξ,η ≤ 1,ξ + η ≤ 1}in the local coordinate system(ξ,η). Under this transformation curved boundary of this triangular element is implicitly replaced by sextic arc. The equation of this arc involves parameters, which are the coordinates of points on the curved side. This paper deduces relations for choosing the parameters in sextic arc in such a way that each arc is always a parabola which passes through four points of the original curve, thus ensuring a good approximation. The point transformations which are thus determined with the above choice of parameters on the curved boundary and also in turn the other parameters in the interior of curved triangle will serve as a powerful subparametric coordinate transformation for higher order curved triangular elements with one curved side and two straight sides.

More »»

2010

Dr. V. Kesavulu Naidu and Dr. K.V. Nagaraja, “The use of parabolic arc in matching curved boundary by point transformations for septic order triangular element and its applications”, Advanced Studies in Contemporary Mathematics (Kyungshang), vol. 20, pp. 437-456, 2010.[Abstract]


This paper is concerned with curved boundary triangular element having one curved side and two straight sides. The curved element considered here is the 36-node (septic) triangular element. On using the isoparametric coordinate transformation, the curved triangle in the global (x,y) coordinate system is mapped into a standard triangle: {(ξ, & etal)/0 ≤ ξ,η ≤ l,ξ+ η ≤ 1} in the local coordinate system (ξ,η). Under this transformation curved boundary of this triangular element is implicitly replaced by septic arc. The equation of this arc involves parameters, which are the coordinates of points on the curved side. This paper deduces relations for choosing the parameters in septic arc in such a way that the arc is always a parabola which passes through four points of the original curve, thus ensuring a good approximation. The point transformations which are thus determined with the above choice of parameters on the curved boundary and also in turn the other parameters inthe interior of curved triangle will serve as a powerful subparametric coordinate transformation for higher order curved triangular elements with one curved side and two straight sides. We have considered an application example, which consists of the quarter ellipse: {(x, y)/x = 0, y = 0, x2/36+y2/4 =1}. We take this as a curved triangle in the physical coordinate system (x, y). We have demonstrated the use of point transformations to determine the points along the curved boundary of the triangle and also the points in the interior of the curved triangle. We have next demonstrated the use of point transformation to determine the arc length of the curved boundary. An additional demonstration which uses the point transformation and the Jacobian is considered. We have thus evaluated certain integrals, for example, ∫/A t αdxdy, (t = x,y,α = 0,1) A and found the physical quantities like area and centroid of the curved triangular elements. We hope that this study gives us the required impetus in the use of higher order curved triangular elements under the subparametric coordinate transformation. More »»

2006

H. T. Rathod, Dr. B. Venkatesh, and Dr. K.V. Nagaraja, “Gauss legendre quadrature formulas over a tetrahedron”, Numerical Methods for Partial Differential Equations, vol. 22, pp. 197–219, 2006.[Abstract]


In this article we consider the Gauss Legendre Quadrature method for numerical integration over the standard tetrahedron: {(x, y, z)|0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in the Cartesian three-dimensional (x, y, z) space. The mathematical transformation from the (x, y, z) space to (ξ, η, ζ) space is described to map the standard tetrahedron in (x, y, z) space to a standard 2-cube: {(ξ, η, ζ)| − 1 ≤ ζ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight coefficients and sampling points. The effectiveness of the formulas is demonstrated by applying them to the integration of three nonpolynomial, three polynomial functions and to the evaluation of integrals for element stiffness matrices in linear three-dimensional elasticity. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

More »»

2005

H. T. Rathod, Dr. B. Venkatesh, and Dr. K.V. Nagaraja, “Gauss Legendre Quadrature Formulae for Tetrahedra”, International Journal for Computational Methods in Engineering Science and Mechanics, vol. 6, pp. 179-186, 2005.[Abstract]


In this paper we consider the Gauss Legendre quadrature method for numerical integration over the standard tetrahedron: (x, y, z)| 0 ≤ x, y, z ≤ 1, x + y + z ≤ 1 in the Cartesian three-dimensional (x, y, z) space. The mathematical transformation from the (x, y, z) space to (ξ, η, ζ) space is described to map the standard tetrahedron in (x, y, z) space to a standard 2-cube: (ξ, η, ζ)| − 1 ≤ ξ, η, ζ ≤ 1 in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight co-efficients and sampling points. The effectiveness of the formulae is demonstrated by applying them to the integration of three nonpolynomial and three polynomial functions.

More »»

2005

H. T. Rathod, Dr. B. Venkatesh, and Dr. K.V. Nagaraja, “Symmetric Gauss Legendre quadrature rules for composite numerical integration over a tetrahedral region”, Journal of Bulletin of Mathematics, vol. 24, pp. 51–79, 2005.

2004

H. Ta Rathod, Dr. K.V. Nagaraja, Dr. B. Venkatesh, and Ramesh, N. Ld, “Gauss Legendre quadrature over a triangle”, Journal of the Indian Institute of Science, vol. 84, pp. 183-188, 2004.[Abstract]


This paper presents a Gauss Legendre quadrature method for numerical integration over the standard triangular surface: {(x, y) | 0 ≤ x, y ≤ 1, x + y ≤ 1} in the Cartesian two-dimensional (x, y) space. Mathematical transformation from (x, y) space to (ξ, η) space map the standard triangle in (x, y) space to a standard 2-square in (ξ, η) space: {(ξ, η)|-l ≤ ξ, η ≤ 1}. This overcomes the difficulties associated with the derivation of new weight coefficients and sampling points and yields results which are accurate and reliable. Results obtained with new formulae are compared with the existing formulae.

More »»

Publication Type: Conference Proceedings

Year of Publication Title

2019

S. Devi, Dr. K.V. Nagaraja, Sarada Jayan, and Smitha, T. V., “2D Higher order triangular mesh generation in irregular domain for finite element analysis using MATLAB”, IOP Conference Series: Materials Science and Engineering, vol. 577. IOP Publishing, p. 012132, 2019.[Abstract]


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.

More »»

2018

T. V. Smitha, Dr. K.V. Nagaraja, and Sarada Jayan, “Automated Mesh Generation Using Curved Cubic Triangular Elements for a Circular Domain with a Finite Element Implementation”, Materials Today: Proceedings, vol. 5. pp. 25203-25211, 2018.[Abstract]


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.

More »»

2017

D. T. Panda, Dr. K.V. Nagaraja, Dr. V. Kesavulu Naidu, and Sarada Jayan, “Application of quintic order parabolic arcs in the analysis of waveguides with arbitrary cross-section”, Proceedings of the International Conference on Communication and Electronics Systems, ICCES 2016. Institute of Electrical and Electronics Engineers Inc., 2017.[Abstract]


A simple and efficient higher order finite element scheme is presented for obtaining highly accurate numerical solution for the two-dimensional Helmholtz equation in waveguides of arbitrary cross-section subjected to dirichlet boundary conditions. The above approach makes use of the Quintic order (5th order) parabolic arcs for accurately mapping the irregular cross section of the waveguide and then transforming the entire waveguide geometry to a standard isosceles triangle. In case of waveguides with regular geometry the transformation is done by straight sided quintic order finite elements. A unique and accurate point transformation technique is developed that ensures high accuracy of mapping by this quintic order curved triangular elements. This point transformation procedure gives a simple interpolating polynomial that defines the transformation from the global coordinate system to the local coordinate system. The above higher order finite element method is found to be highly optimal and accurate considering the various computational parameters like the number of triangular elements, degrees of freedom, nodal point distribution on the entire geometry, etc. © 2016 IEEE.

More »»

Publication Type: Book Chapter

Year of Publication Title

2018

T. Darshi Panda, Dr. K.V. Nagaraja, and Dr. V. Kesavulu Naidu, “A Simple and Efficient Higher Order Finite Element Scheme for Helmholtz Waveguides”, in Advances in Electronics, Communication and Computing, vol. 443, A. Kalam, Das, S., and Sharma, K., Eds. Singapore: Springer Singapore, 2018, pp. 421-43.[Abstract]


This paper presents a simple and efficient finite element scheme for computing the cutoff wave numbers of arbitrary-shaped waveguides using higher order triangular elements. The waveguide geometry is divided into a set of triangular elements and each of these elements is mapped to a standard isosceles triangle by discritizing with subparametric finite elements. For waveguides containing arbitrary cross sections, the transformation is done using a series of higher order parabolic arcs. In this case, the curve boundaries are approximated by curved triangular finite elements and then transformed to an isosceles triangle. Numerical results are illustrated to validate the present approach. The obtained results have converged very well with the existing literature with minimum number of triangular elements, degree of freedoms, order of computational matrix, etc.

More »»

Research Grants Received

Year  Funding Agency Title of the Project Investigators Status
2017 National Board of Higher Mathematics (NBHM) Application of Sub-parametric Finite Elements for Eigenanalysis of Waveguides in Electromagnetics Dr. K.V. Nagaraja Completed

Invited Talks

  • “Introduction to Finite Element Method” on August 1, 2014, in a “One-Week Faculty Development Program”, Organized by the Department of Mathematics, Malnad College of Engineering, Hassan, from July 28- August 2, 2014. The workshop was sponsored by TEQIP-II (Technical Education Quality Improvement Programme-II).
  • “Finite element solution of Darcy-Brinkman and Darcy-Forchheimer-Brinkman equation for some flow channels using triangular elements” on June 27, 2014, in a “Two-Week International Workshop on Computational Fluid Dynamics” Organized by the Department of Mathematics, BMS College of Engineering, Bangalore, during June 23- July 5, 2014. The workshop was sponsored by TEQIP-II (Technical Education Quality Improvement Programme-II).
  • “Finite Element Method and its Applications”, Two-day workshop on “Current Topics in Mathematics”, for Post Graduate Students on March 1, 2013, at Christ University.
  • “An overview of Elementary Calculus”, At DST- INSPIRE INTERNSHIP SCIENCE CAMP organized by Division of Biological Sciences, School of Natural Science, Bangalore University on May 17, 2011.

Editorial Board Member

  • International Journal of Mathematics and Computer Applications Research
  • Universal Journal of Applied Mathematics (Horizon Research Publishing (HRPUB), USA).
  • Journal of Scientific Research in Physical and Mathematical Sciences
  • International Journal of Multidisciplinary Research and Modern Education

Reviewer

International Journals

  • Journal of Computational and Applied Mathematics (Elsevier)
  • American Mathematical Society
  • Computer Methods in Applied Mechanics and Engineering (Elsevier)
  • International Journal for Computational Methods in Engineering Science and Mechanics (Taylor & Francis)
  • The Scientific World Journal (Hindawi Publishing Corporation)
  • Universal Journal of Applied Mathematics
  • Journal of Applied Mathematics (Hindawi Publishing Corporation)

Courses Taught

  • Optimization Techniques
  • Numerical Methods
  • Finite Element Method
  • Single Variable Calculus
  • Multi-Variable Calculus
  • Integral Transforms
  • Ordinary Differential Eqs and Partial Differential Eqs.
  • Probability and Statistics
  • Advanced Numerical Analysis

Ph.D. Student Guidance 

Sl. No. Name of the Student(s) Topic Status – Ongoing /Completed Year of Completion
1. Padmasudha Kannan Application of Sub-parametric Finite Elements for Eigenanalysis of Waveguides in Electromagnetics Ongoing  
2. Supriya Devi Investigation of The Flow Behaviour Over NACA Airfoils Using Subparametric Finite Element Method Completed 2020
3. T. V. Smitha Automating the Higher-Order Finite Element Method Using the Subparametric Transformations for Elliptic Partial Differential Equations Completed 2019
4. Sarada Jayan Effective Numerical Integration Methods to evaluate Multiple integrals using Generalized Gaussian Quadrature Completed 2014
5. V. Kesavulu Naidu Finite Element Method solution of Partial Differential Equations using the Subparametric Transformations for some Higher Order Triangular Elements. Completed 2013