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

Dr. K. V. Nagaraja currently serves as Professor at department of Mathematics, Amrita School of Engineering, Banglore campus. He has a teaching experience of 19 years and research experience of 16 years. He has guided 2 PhD students and currently guiding 4 PhD students.

Qualification

Degree Name of University Year
M.Sc. Bangalore University 1996
M.Phil. Bangalore University 1998
PhD Bangalore University 2005

Publications

Publication Type: Conference Paper

Year of Publication Publication Type Title

2017

Conference Paper

D. T. Panda, Dr. K.V. Nagaraja, V. Naidu, K., and Sarada Jayan, “Application of quintic order parabolic arcs in the analysis of waveguides with arbitrary cross-section”, in Proceedings of the International Conference on Communication and Electronics Systems, ICCES 2016, 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: Journal Article

Year of Publication Publication Type Title

2016

Journal Article

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 »»

2015

Journal Article

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.

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2015

Journal Article

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.

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2015

Journal Article

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.

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2015

Journal Article

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

Journal Article

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.

2013

Journal Article

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.

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2012

Journal Article

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>

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2011

Journal Article

H. Ta Rathod, Venkatesh, Bb, 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.

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2011

Journal Article

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

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2011

Journal Article

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.

2010

Journal Article

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 »»

2010

Journal Article

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

Journal Article

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

Journal Article

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 »»

2008

Journal Article

H. Ta Rathod, Dr. K.V. Nagaraja, Dr. V. Kesavulu Naidu, and Venkatesudu, Bb, “The use of parabolic arcs in matching curved boundaries by point transformations for some higher order triangular elements”, Finite Elements in Analysis and Design, vol. 44, pp. 920-932, 2008.[Abstract]


This paper is concerned with curved boundary triangular elements having one curved side and two straight sides. The curved elements considered here are the 6-node (quadratic), 10-node (cubic), 15-node (quartic) and 21-node (quintic) triangular elements. On using the isoparametric coordinate transformation, these curved triangles in the global (x, y) coordinate system are 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 these triangular elements is implicitly replaced by quadratic, cubic, quartic and quintic arcs. The equations of these arcs involve parameters, which are the coordinates of points on the curved side. This paper deduces relations for choosing the parameters in quartic and quintic arcs 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 triangles will serve as a powerful subparametric coordinate transformation for higher order curved triangular elements with one curved side and two straight sides. © 2008 Elsevier B.V. All rights reserved.

More »»

2005

Journal Article

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, no. 3, 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 »»

2004

Journal Article

H. Ta Rathod, Dr. K.V. Nagaraja, Venkatesudu, Bc, 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 »»

Invited Talks/Workshops Attended

  1. Dr. K. V. Nagaraja has delivered  an invited talk titled on “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.
  2. Dr. K. V. Nagaraja has delivered  an invited talk titled on “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).
  3. Dr. K. V. Nagaraja has delivered an invited talk titled on “Introduction to Finite Element Method” on August 1, 2014 in a “One-Week Faculty Development Programmee” Organized by the Department of Mathematics, Malnad College of Engineering, Hassan during July 28 – August 2, 2014. The workshop was sponsored by TEQIP-II (Technical Education Quality Improvement Programme-II).
  4. Dr. K. V. Nagaraja , attended National Workshop on Parabolic Differential Equations and Applications to Image Processing during October 26 - 29, 2015, at Sri Sathya Sai Institute of Higher Learning, Puttaparti.

PhD Students

  1. Mr. V. Kesavulu Naidu

    Area of Research: Finite Element Method
    Viva-Voce conducted on November 8, 2013
    Title: Finite Element Method solution of Partial Differential Equations using the Subparametric Transformations for some Higher Order Triangular Elements.

  2. Ms. Sarada Jayan

    Area of Research: Numerical Analysis
    Viva-Voce conducted on December 16, 2014
    Title: Effective Numerical Integration Methods to evaluate Multiple integrals using Generalized Gaussian Quadrature.

  3. Ms. Supriya Devi (Part Time)

    Department of Mathematics
    ASE, Bangalore
    Area of Research: Finite Element Method
    Registered on March 2015

  4. Ms. Smitha TV(Full Time)

    Department of Mathematics
    ASE, Bangalore
    Area of Research: Finite Element Method
    Registered on September 2015

  5. Padma Sudha(Part Time)

    Department of Mathematics
    ASE, Bangalore
    Area of Research: Numerical Optimization Techniques
    Registered on September 2015

  6. Ms. Sandhya Rani(Part Time)

    Department of Mathematics
    ASE, Bangalore
    Area of Research: Numerical Optimization Techniques
    Registered on September 2015
     

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