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: Journal Article

Year of Publication

Publication Type

Title

2015

Journal Article

J. Sarada and .V.Nagaraja, K., “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.

VaKesavulu Naidu, Siddheshwar, P. G., and .V.Nagaraja, K., “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.

J. Sarada and .V.Nagaraja, K., “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.

J. Sarada and .V.Nagaraja, K., “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 .V.Nagaraja, K., “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

VaKesavulu Naidu and .V.Nagaraja, K., “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]

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

H. Ta Rathod, Venkatesh, B., and .V.Nagaraja, K., “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.

J. Sarada and .V.Nagaraja, K., “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

H. .T.Rathod, .V.Nagaraja, K., 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

VaKesavulu Naidu and .V.Nagaraja, K., “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
More »»

2010

Journal Article

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

Journal Article

K. .V.Nagaraja, Naidu, VaKesavulu, 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.

H. .T.Rathod, Gali, A., Shivaram, K. T., and .V.Nagaraja, K., “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., .V.Nagaraja, K., and Naidu, VaKesavulu, “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, .V.Nagaraja, K., Naidu, VaKesavulu, and Venkatesh, B., “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]

H. T. Rathod, .V.Nagaraja, K., and Venkatesh, B., “On the application of two symmetric Gauss Legendre quadrature rules for composite numerical integration over a triangular surface”, Applied mathematics and computation, vol. 190, pp. 21–39, 2007.[Abstract]

This paper first presents a Gauss Legendre quadrature rule for the evaluation of View the MathML source, where f(x,y) is an analytic function in x, y and T is the standard triangular surface: {(x,y)|0⩽x,y⩽1,x+y⩽1} in the two space (x,y). We transform this integral into an equivalent integral View the MathML source where S is the 2-square in (ξ, η ) space: {(ξ,η)|-1⩽ξ,η⩽1}. We then apply the one-dimensional Gauss Legendre quadrature rules in ξ and η variables to arrive at an efficient Quadrature rules with new weight coefficients and new sampling points. Then a second Gauss Legendre quadrature rule of composite type is obtained. This rule is derived by discretising T into three new triangles View the MathML source of equal size which are obtained by joining centroid of T , C=(1/3,1/3) to the three vertices of T . By use of affine transformations defined over each View the MathML source and the linearity property of integrals leads to the result:
More »»

2007

Journal Article

H. T. Rathod, .V.Nagaraja, K., and Venkatesh, B., “Numerical integration of some functions over an arbitrary linear tetrahedra in Euclidean three-dimensional space”, Applied Mathematics and Computation, vol. 191, pp. 397–409, 2007.[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. We have applied Gauss Legendre Quadrature rules which were recently derived by Rathod et al. [H.T. Rathod, K.V. Nagaraja, B. Venkatesudu, N.L. Ramesh, Gauss Legendre Quadrature over a Triangle, J. Indian Inst. Sci. 84 (2004) 183–188] to evaluate the typical integrals governed by the proposed method.
More »»

2007

Journal Article

H. T. Rathod, Venkatesh, B., and .V.Nagaraja, K., “On the application of two Gauss–Legendre quadrature rules for composite numerical integration over a tetrahedral region”, Applied mathematics and computation, vol. 189, pp. 131–162, 2007.[Abstract]

In this paper we first present a Gauss–Legendre quadrature rule for the evaluation of View the MathML source, where f(x, y, z) is an analytic function in x, y, z and T is the standard tetrahedral region: {(x, y, z)∣0 ⩽ x, y, z ⩽ 1, x + y + z ⩽ 1} in three space (x, y, z). We then use a transformation x = x(ξ, η, ζ), y = y(ξ, η, ζ) and z = z(ξ, η, ζ ) to change the integral into an equivalent integral View the MathML source over the standard 2-cube in (ξ, η, ζ) space: {(ξ, η, ζ)∣ −1 ⩽ ξ, η, ζ ⩽ 1}. We then apply the one-dimensional Gauss–Legendre quadrature rules in ξ, η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss–Legendre quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra View the MathML source (i = 1, 2, 3, 4) of equal size which are obtained by joining the centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T . By use of the affine transformations defined over each View the MathML source and the linearity property of integrals leads to the result:
More »»

2006

Journal Article

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

H. T. Rathod, Venkatesh, B., and .V.Nagaraja, K., “On the application of two symmetric Gauss Legendre quadrature rules for composite numerical integration over a tetrahedral region”, International Journal for Computational Methods in Engineering Science and Mechanics, vol. 7, no. 6, pp. 445–459, 2006.[Abstract]

In this paper, we first present a Gauss Legendre Quadrature rule for the evaluation of I = ∫∫∫T f(x,y,z) dxdydz , where f(x,y,z) is an analytic function in x,y,z and T is the standard tetrahedral region: {(x,y,z) |0 ≤ x,y,z ≤ 1,x + y + z ≤ 1} in three space ( x,y,z) . We then use the transformations x = x(ξ,η,ζ), y = y (ξ,η,ζ) and z = z(ξ,η,ζ) to change the integral I into an equivalent integral I = ∫− 1 1∫− 1 1∫− 1 1f(x(ξ,η,ζ),y(ξ,η,ζ),z(ξ,η,ζ)) dξ dη dζ over the standard 2-cube in (ξ,η,ζ) space: {(ξ,η,ζ) | − 1 ≤ ξ,η,ζ ≤ 1} . We then apply the one-dimensional Gauss Legendre Quadrature rule in ξ , η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss Legendre Quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra Ti c ( i = 1,2,3,4) of equal size, which are obtained by joining centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T. Use of the affine transformations defined over each Ti c and the linearity property of integrals leads to the result:
More »»

2005

Journal Article

H. T. Rathod, Venkatesh, B., and .V.Nagaraja, K., “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 »»

Invited Talks/Workshops Attended

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.

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).

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).

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

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.

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.

Ms. Supriya Devi (Part Time)

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

Ms. Smitha TV(Full Time)

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

Padma Sudha(Part Time)

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

Ms. Sandhya Rani(Part Time)

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

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

## Publications

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 »»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 »»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 »»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 »»<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 »»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 »»{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 »»{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 More »»

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 »»

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 »»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 »»

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 »»This paper first presents a Gauss Legendre quadrature rule for the evaluation of View the MathML source, where f(x,y) is an analytic function in x, y and T is the standard triangular surface: {(x,y)|0⩽x,y⩽1,x+y⩽1} in the two space (x,y). We transform this integral into an equivalent integral View the MathML source where S is the 2-square in (ξ, η ) space: {(ξ,η)|-1⩽ξ,η⩽1}. We then apply the one-dimensional Gauss Legendre quadrature rules in ξ and η variables to arrive at an efficient Quadrature rules with new weight coefficients and new sampling points. Then a second Gauss Legendre quadrature rule of composite type is obtained. This rule is derived by discretising T into three new triangles View the MathML source of equal size which are obtained by joining centroid of T , C=(1/3,1/3) to the three vertices of T . By use of affine transformations defined over each View the MathML source and the linearity property of integrals leads to the result: More »»

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. We have applied Gauss Legendre Quadrature rules which were recently derived by Rathod et al. [H.T. Rathod, K.V. Nagaraja, B. Venkatesudu, N.L. Ramesh, Gauss Legendre Quadrature over a Triangle, J. Indian Inst. Sci. 84 (2004) 183–188] to evaluate the typical integrals governed by the proposed method. More »»

In this paper we first present a Gauss–Legendre quadrature rule for the evaluation of View the MathML source, where f(x, y, z) is an analytic function in x, y, z and T is the standard tetrahedral region: {(x, y, z)∣0 ⩽ x, y, z ⩽ 1, x + y + z ⩽ 1} in three space (x, y, z). We then use a transformation x = x(ξ, η, ζ), y = y(ξ, η, ζ) and z = z(ξ, η, ζ ) to change the integral into an equivalent integral View the MathML source over the standard 2-cube in (ξ, η, ζ) space: {(ξ, η, ζ)∣ −1 ⩽ ξ, η, ζ ⩽ 1}. We then apply the one-dimensional Gauss–Legendre quadrature rules in ξ, η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss–Legendre quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra View the MathML source (i = 1, 2, 3, 4) of equal size which are obtained by joining the centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T . By use of the affine transformations defined over each View the MathML source and the linearity property of integrals leads to the result: More »»

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 »»

In this paper, we first present a Gauss Legendre Quadrature rule for the evaluation of I = ∫∫∫T f(x,y,z) dxdydz , where f(x,y,z) is an analytic function in x,y,z and T is the standard tetrahedral region: {(x,y,z) |0 ≤ x,y,z ≤ 1,x + y + z ≤ 1} in three space ( x,y,z) . We then use the transformations x = x(ξ,η,ζ), y = y (ξ,η,ζ) and z = z(ξ,η,ζ) to change the integral I into an equivalent integral I = ∫− 1 1∫− 1 1∫− 1 1f(x(ξ,η,ζ),y(ξ,η,ζ),z(ξ,η,ζ)) dξ dη dζ over the standard 2-cube in (ξ,η,ζ) space: {(ξ,η,ζ) | − 1 ≤ ξ,η,ζ ≤ 1} . We then apply the one-dimensional Gauss Legendre Quadrature rule in ξ , η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss Legendre Quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra Ti c ( i = 1,2,3,4) of equal size, which are obtained by joining centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T. Use of the affine transformations defined over each Ti c and the linearity property of integrals leads to the result: More »»

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 »»

Invited Talks/Workshops Attended

## PhD Students

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.

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.

Department of Mathematics

ASE, Bangalore

Area of Research: Finite Element Method

Registered on March 2015

Department of Mathematics

ASE, Bangalore

Area of Research: Finite Element Method

Registered on September 2015

Department of Mathematics

ASE, Bangalore

Area of Research: Numerical Optimization Techniques

Registered on September 2015

Department of Mathematics

ASE, Bangalore

Area of Research: Numerical Optimization Techniques

Registered on September 2015

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