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E-book: Finite Element Method for Boundary Value Problems: Mathematics and Computations [Taylor & Francis e-book]

(Texas A&M University, College Station, USA), (University of Kansas, USA)
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Written by two well-respected experts in the field, The Finite Element Method for Boundary Value Problems: Mathematics and Computations bridges the gap between applied mathematics and application-oriented computational studies using FEM. Mathematically rigorous, the FEM is presented as a method of approximation for differential operators that are mathematically classified as self-adjoint, non-self-adjoint, and non-linear, thus addressing totality of all BVPs in various areas of engineering, applied mathematics, and physical sciences. These classes of operators are utilized in various methods of approximation: Galerkin method, Petrov-Galerkin Method, weighted residual method, Galerkin method with weak form, least squares method based on residual functional, etc. to establish unconditionally stable finite element computational processes using calculus of variations. Readers are able to grasp the mathematical foundation of finite element method as well as its versatility of applications. h-, p-, and k-versions of finite element method, hierarchical approximations, convergence, error estimation, error computation, and adaptivity are additional significant aspects of this book.
Preface xix
About the Authors xxv
1 Introduction
1(14)
1.1 General Comments and Basic Philosophy
1(2)
1.2 Basic Concepts of the Finite Element Method
3(10)
1.2.1 Discretization
3(2)
1.2.2 Local approximation
5(2)
1.2.3 Integral forms and algebraic equations over an element
7(1)
1.2.4 Assembly of element equations
8(1)
1.2.5 Computation of the solution
9(1)
1.2.6 Post-processing
9(1)
1.2.7 Remarks
9(2)
1.2.8 k-version of the finite element method and hpk framework
11(2)
1.3 Summary
13(2)
2 Concepts from Functional Analysis
15(54)
2.1 General Comments
15(1)
2.2 Sets, Spaces, Functions, Functions Spaces, and Operators
15(16)
2.2.1 Hilbert spaces Hk(Ω)
18(1)
2.2.2 Definition of scalar product in Hk(Ω) space
19(1)
2.2.3 Properties of scalar product
19(1)
2.2.4 Norm of u in Hilbert space Hk(Ω)
20(1)
2.2.5 Seminorm of u in Hilbert space Hk(Ω)
20(2)
2.2.6 Function spaces
22(1)
2.2.7 Operators
23(1)
2.2.8 Types of operators
24(3)
2.2.9 Energy product
27(1)
2.2.10 Integration by parts (IBP)
27(4)
2.3 Elements of Calculus of Variations
31(13)
2.3.1 Concept of the variation of a functional
33(1)
2.3.2 Euler's equation: Simplest variational problem
34(7)
2.3.3 Variation of a functional: some practical aspects
41(1)
2.3.4 Riemann and Lebesgue integrals
42(2)
2.4 Examples of Differential Operators and their Properties
44(20)
2.4.1 Self-adjoint differential operators
44(14)
2.4.2 Non-self-adjoint differential operators
58(5)
2.4.3 Non-linear differential operators
63(1)
2.5 Summary
64(5)
3 Classical Methods of Approximation
69(124)
3.1 Introduction
69(1)
3.2 Basic Steps in Classical Methods of Approximation based on Integral Forms
70(2)
3.3 Integral forms using the Fundamental Lemma of the Calculus of Variations
72(23)
3.3.1 The Galerkin method
73(1)
3.3.1.1 Self-adjoint and non-self-adjoint linear differential operators
74(3)
3.3.1.2 Non-linear differential operators
77(1)
3.3.2 The Petrov-Galerkin and weighted-residual methods
78(1)
3.3.2.1 Self-adjoint and non-self-adjoint linear differential operators
78(2)
3.3.2.2 Non-linear differential operators
80(1)
3.3.3 The Galerkin method with weak form
81(4)
3.3.3.1 Linear differential operators
85(1)
3.3.3.2 Non-linear differential operators
86(1)
3.3.4 The least-squares method
87(5)
3.3.4.1 Self-adjoint and non-self-adjoint linear differential operators
92(1)
3.3.4.2 Non-linear differential operators
93(1)
3.3.5 Collocation method
94(1)
3.4 Approximation Spaces for Various Methods of Approximation
95(2)
3.5 Integral Formulations of BVPs using the Classical Methods of Approximations
97(56)
3.5.1 Self-adjoint differential operators
98(29)
3.5.2 Non-self-adjoint Differential Operators
127(14)
3.5.3 Non-linear Differential Operators
141(12)
3.6 Numerical Examples
153(32)
3.7 Summary
185(8)
4 The Finite Element Method
193(30)
4.1 Introduction
193(1)
4.2 Basic steps in the finite element method
194(27)
4.2.1 Discretization
194(4)
4.2.2 Construction of integral forms over an element
198(1)
4.2.2.1 Integral forms for GM, PGM, and WRM
199(1)
4.2.2.2 Integral form for GM/WF
200(2)
4.2.2.3 Integral form based on residual functional
202(1)
4.2.3 The local approximation φeh of φ over an element
203(1)
4.2.4 Element matrices and vectors resulting from the integral form and the local approximation
204(1)
4.2.4.1 Galerkin method, Petrov--Galerkin method, and weighted residual method
205(1)
4.2.4.2 Galerkin method with weak form
206(4)
4.2.4.3 Least-squares process based on residual functional
210(2)
4.2.5 Assembly of element equations: GM, PGM, WRM, GM/WF and LSP when A is linear
212(2)
4.2.6 Consideration of boundary conditions in the assembled equations
214(1)
4.2.6.1 GM, PGM, WRM, and LSP based on the residual functional
214(1)
4.2.6.2 GM/WF
215(1)
4.2.7 Computation of the solution: finite element processes based on all methods of approximation except LSP for non-linear operators
216(1)
4.2.8 Assembly of element equations and their solution in finite element processes based on residual functional (LSP) when the differential operator A is non-linear
217(3)
4.2.9 Post processing of the solution
220(1)
4.3 Summary
221(2)
5 Self-Adjoint Differential Operators
223(140)
5.1 Introduction
223(3)
5.1.1 GM/WF
224(1)
5.1.2 LSP based on residual functional
225(1)
5.2 One-dimensional BVPs in a single dependent variable
226(84)
5.2.1 ID steady-state diffusion equation: finite element processes based on GM/WF
226(1)
5.2.1.1 Discretization
227(1)
5.2.1.2 Integral form using GM/WF (weak form) of the BVP for an element e with domain Ωe
227(4)
5.2.1.3 Approximation space Vh, test function space V and local approximation φeh
231(2)
5.2.1.4 Local approximation φeh and mapping Ωe to Ωε
233(1)
5.2.1.5 Element equations
234(1)
5.2.1.6 Assembly of element equations and computation of the solution
235(2)
5.2.1.7 Inter-element continuity conditions on PVs or dependent variables
237(1)
5.2.1.8 Rules for assembling element matrices and vectors
237(5)
5.2.1.9 Inter-element continuity conditions on the sum of secondary variables
242(1)
5.2.1.10 Imposition of EBCs
243(1)
5.2.1.11 Solving for unknown degrees of freedom
243(1)
5.2.1.12 Special case: numerical study
244(2)
5.2.1.13 Post-processing of solution
246(1)
5.2.1.14 Analytical solution and comparison with finite element solutions
246(5)
5.2.2 1D steady-state diffusion equation
251(1)
5.2.2.1 Case (a): a = 1, L = 1, q(x) = xn, n is a positive integer
252(8)
5.2.2.2 Case (b): a = 1, L = 1, q(x) = sin nπx, n = 4
260(5)
5.2.3 Least-squares finite element formulation
265(10)
5.2.3.1 Approximation space Vh
275(1)
5.2.3.2 Numerical studies
275(3)
5.2.4 LSFEP using auxiliary variables and auxiliary equations
278(7)
5.2.4.1 Approximation spaces for φeh and τeh
285(1)
5.2.4.2 Numerical studies
285(4)
5.2.5 One-dimensional heat conduction with convective boundary
289(7)
5.2.5.1 Approximation space Vh
296(1)
5.2.5.2 Numerical study
297(3)
5.2.6 1D axisymmetric heat conduction
300(1)
5.2.6.1 Galerkin method with weak form
301(2)
5.2.6.2 LSM based on residual functional
303(2)
5.2.7 A 1D BVP governed by a fourth-order differential operator
305(4)
5.2.7.1 Approximation space Vh
309(1)
5.3 Two-dimensional boundary value problems
310(34)
5.3.1 A general 2D BVP in a single dependent variable
310(4)
5.3.1.1 Definition of Cle: element geometry
314(2)
5.3.1.2 Approximation space Vh
316(1)
5.3.1.3 Computation of the element matrix [ Ke] and vector {Fe}
317(1)
5.3.1.4 Details of secondary variable vector {Pe}
317(3)
5.3.2 2D Poisson's equation: numerical studies
320(1)
5.3.2.1 Case (a): f = 1 with BCs φ(±1, y) = φ(x, ±1) = 0; GM/WF
320(7)
5.3.2.2 Case (b): BCs φ(±1, y) = φ(x, ±1) = 1.0; GM/WF
327(1)
5.3.3 Two-dimensional boundary value problems in multi-variables: 2D plane elasticity
328(4)
5.3.3.1 Galerkin method with weak form
332(5)
5.3.3.2 Least-squares method using residual functional
337(7)
5.4 Three-dimensional boundary value problems
344(14)
5.4.1 Three-dimensional boundary value problems in a single dependent variable
344(1)
5.4.1.1 Galerkin method with weak form
345(2)
5.4.1.2 Approximation space
347(1)
5.4.1.3 Local approximation Teh
347(1)
5.4.1.4 Definition of Ωe: Element geometry
348(2)
5.4.1.5 Computations of element matrix [ Ke] and vector {Fe}
350(1)
5.4.1.6 Details of secondary variable vector {Pe}
350(3)
5.4.2 Three-dimensional boundary value problems in multivariables
353(2)
5.4.2.1 Galerkin method with weak form
355(2)
5.4.2.2 Approximation spaces
357(1)
5.4.2.3 Local approximation
357(1)
5.5 Summary
358(5)
6 Non-Self-Adjoint Differential Operators
363(56)
6.1 Introduction
363(2)
6.2 1D convection-diffusion equation
365(25)
6.2.1 Analytical solution
365(2)
6.2.2 The Galerkin method with weak form (GM/WF)
367(11)
6.2.3 Least squares finite element formulation
378(2)
6.2.4 Least squares formulation: first order system
380(10)
6.3 2D convection-diffusion equation
390(23)
6.3.1 Least squares finite element formulation based on the residual functional
395(2)
6.3.2 Least squares finite element formulation of (6.113) by recasting it as a system of first order PDEs
397(4)
6.3.3 Convection dominated thermal flow (advection skewed to a square domain)
401(7)
6.3.4 Advection of a cosine hill in a rotating flow field
408(3)
6.3.5 Thermal boundary layer
411(2)
6.4 Summary
413(6)
7 Non-Linear Differential Operators
419(74)
7.1 Introduction
419(3)
7.2 One dimensional Burgers equation
422(20)
7.2.1 The Galerkin method with weak form
423(5)
7.2.2 LSP based on residual functional
428(1)
7.2.3 LSP based on residual functional: first order system of equations
429(13)
7.3 Fully developed flow of Giesekus fluid between parallel plates (polymer flow)
442(8)
7.4 2D steady-state Navier--Stokes equations
450(21)
7.4.1 LSP based on residual functional: first order system of PDEs
452(2)
7.4.2 LSP based on residual functional: higher order systems of PDEs
454(1)
7.4.3 Slider bearing; flow of a viscous lubricant
455(2)
7.4.4 A square lid-driven cavity
457(4)
7.4.5 Asymmetric backward facing step
461(6)
7.4.6 Flow past a circular cylinder
467(4)
7.5 2D compressible Newtonian fluid flow
471(15)
7.5.1 Carter's plate
474(2)
7.5.1.1 Mach 1 flow
476(2)
7.5.1.2 General consideration for higher Mach number flows
478(1)
7.5.1.3 Mach 2 flow
479(1)
7.5.1.4 Mach 3 flow
480(1)
7.5.1.5 Mach 5 flow
481(2)
7.5.2 Mach 1 flow past a circular cylinder
483(3)
7.6 Summary
486(7)
8 Basic Elements of Mapping and Interpolation Theory
493(116)
8.1 Mapping in one dimension
493(2)
8.1.1 Mapping of points
494(1)
8.1.2 Mapping of lengths
494(1)
8.1.3 Behavior of dependent variable φ over Ωe
495(1)
8.2 Elements of interpolation theory over Ωε = [ -- 1, 1]
495(21)
8.2.1 A polynomial approximation in one dimension
495(2)
8.2.2 Lagrange interpolating polynomials in one dimension
497(4)
8.2.3 p-version hierarchical functions in one dimension
501(6)
8.2.4 Higher order global differentiability approximations in one dimension: p-version
507(1)
8.2.4.1 Local approximation of class C1(Ωe)
508(3)
8.2.4.2 Interpolations or local approximations of class C2(Ωe
511(4)
8.2.4.3 Local approximations of class C1(Ωe)
515(1)
8.3 Mapping in two dimensions: quadrilateral elements
516(4)
8.4 Local approximation over Ωm: quadrilateral elements
520(12)
8.4.1 C00 local approximations over Ωεη: polynomial approach
522(3)
8.4.2 C00 Lagrange type local approximation using tensor product
525(4)
8.4.3 C00 p-version hierarchical local approximations based on Lagrange polynomials
529(3)
8.5 2D Cij(Ωe) p-version local approximations
532(10)
8.5.1 2D interpolations of type C11(Ωe) with p-levels of and pε and pn
538(2)
8.5.2 2D interpolations of type C22(Ωe) with p-levels of pεand pn
540(2)
8.5.3 2D Cij(Ωe) interpolations of p-levels pε and pη
542(1)
8.6 2D Cij(Ωe) approximations for quadrilateral elements
542(14)
8.6.1 C11 HGDA for 2D distorted quadrilateral elements in xy space
549(1)
8.6.2 C22 HGDA for 2D distorted quadrilateral elements in xy space
550(1)
8.6.3 C33 HGDA for 2D distorted quadrilateral elements in xy space
551(2)
8.6.4 Derivation of Cij approximations for distorted quadrilateral elements
553(1)
8.6.5 Limitations of 2D C11 global differentiability local approximations for distorted quadrilateral elements
554(2)
8.7 Interpolation theory for 2D triangular elements
556(10)
8.7.1 Langrange family C00 basis functions based on Pascal triangle
556(2)
8.7.2 Lagrange family C00 basis functions based on area coordinates
558(1)
8.7.3 Higher degree C00 basis functions using area coordinates
559(7)
8.8 1D and 2D approximations based on Legendre polynomials
566(11)
8.8.1 Legendre polynomials
566(1)
8.8.2 1D p-version C0 hierarchical approximation functions (Legendre polynomials)
567(1)
8.8.3 2D p-version C00 hierarchical interpolation functions for quadrilateral elements (Legendre polynomials)
567(1)
8.8.4 2D Cij p-version interpolations functions for quadrilateral elements (Legendre polynomials)
568(1)
8.8.5 2D C00 p-version interpolation functions for triangular elements (Legendre polynomials)
568(3)
8.8.6 2D Cij interpolation functions for triangular elements (Legendre polynomials)
571(6)
8.9 1D and 2D interpolations based on Chebyshev polynomials
577(1)
8.9.1 Chebyshev polynomials
577(1)
8.9.2 1D C0 p-version hierarchical interpolations based on Chebyshev polynomials
577(1)
8.9.3 2D p-version C00 hierarchical interpolation functions for quadrilateral elements (Chebyshev polynomials)
578(1)
8.9.4 2D Cij p-version interpolation functions for quadrilateral elements (Chebyshev polynomials)
578(1)
8.10 Serendipity family of C00 interpolations
578(6)
8.10.1 Method of deriving serendipity interpolation functions
579(5)
8.11 Interpolation functions for 3D elements
584(20)
8.11.1 Hexahedron elements
584(1)
8.11.1.1 Mapping of points
584(2)
8.11.1.2 Mapping of lengths
586(1)
8.11.1.3 Mapping of volumes
586(1)
8.11.1.4 Obtaining derivatives of φeh(ε, η σ) with respect to x, y, z
587(1)
8.11.2 Local approximation for a dependent variable φover Ωm
588(1)
8.11.2.1 Hexahedron elements
588(2)
8.11.2.2 Higher degree approximations of φ over Ωm
590(2)
8.11.2.3 C000 Lagrange type local approximations using tensor product
592(5)
8.11.2.4 C000 p-version 3D hierarchical local approximations: using tensor product
597(2)
8.11.2.5 3D Cijk(Ωe) p-version local approximations: Hexahedron elements
599(1)
8.11.2.6 3D Cijk(Ωe) p-version interpolations for distorted hexahedron elements: 27 node element
600(1)
8.11.2.7 Interpolation theory for 3D tetrahedron elements: basis functions of class C000(Ωe) based on Lagrange interpolations
600(1)
8.11.2.8 Lagrange family C000 interpolations based on volume coordinates
601(2)
8.11.2.9 Higher degree C000 basis functions using volume coordinates
603(1)
8.11.2.10 Four-node linear tetrahedron element (p-level of one)
604(1)
8.11.2.11 A ten-node tetrahedron element (p-level of 2)
604(1)
8.12 Summary
604(5)
9 Linear Elasticity using the Principle of Minimum Total Potential Energy
609(16)
9.1 Introduction
609(1)
9.2 New notation
610(1)
9.3 Approach
611(1)
9.4 Element equations
611(6)
9.4.1 Local approximation of the displacement field
611(1)
9.4.2 Stresses and strains
612(1)
9.4.3 Strain energy πe1 and potential energy of loads πe2
612(2)
9.4.4 Total potential energy πe for an element e
614(3)
9.5 Finite element formulation for 2D linear elasticity
617(7)
9.5.1 Local approximation of u and v over Ωe or Ωεη
618(1)
9.5.2 Stresses, strains and constitutive equations
618(1)
9.5.3 [ B] matrix relating strains to nodal degrees of freedom
619(1)
9.5.4 Element stiffness matrix [ Ke]
619(1)
9.5.5 Transformations from (ε, η) to (x, y) space
620(1)
9.5.6 Body forces
620(1)
9.5.7 Initial strains (thermal loads)
621(1)
9.5.8 Equivalent nodal loads {Fe}p due to pressure acting normal to the element faces
622(2)
9.6 Summary
624(1)
10 Linear and Nonlinear Solid Mechanics using the Principle of Virtual Displacements
625(26)
10.1 Introduction
625(1)
10.2 Principle of virtual displacements
626(1)
10.3 Virtual work statements
627(8)
10.3.1 Stiffness matrix
633(2)
10.4 Solution method
635(2)
10.4.1 Summary of solution procedure
636(1)
10.5 Finite element formulation for 2D solid continua
637(4)
10.6 Finite element formulation for 3D solid continua
641(3)
10.7 Axisymmetric solid finite elements
644(4)
10.8 Summary
648(3)
11 Additional Topics in Linear Structural Mechanics
651(32)
11.1 Introduction
651(1)
11.2 1D axial spar or rod element in R1 (1D space)
651(5)
11.2.1 Stresses and strains
653(1)
11.2.2 Total potential energy: πe
653(3)
11.3 1D axial spar or rod element in R2
656(10)
11.3.1 Coordinate transformation
656(3)
11.3.2 A two member truss
659(1)
11.3.2.1 Computations
660(5)
11.3.2.2 Post-processing
665(1)
11.4 1D axial spar or rod element in R3 (3D space)
666(2)
11.5 The Euler--Bernoulli beam element
668(7)
11.5.1 Derivation of the element equations (GM/WF)
670(1)
11.5.2 Local approximation
671(4)
11.6 Euler-Bernoulli frame elements in R2
675(2)
11.7 The Timoshenko beam elements
677(4)
11.7.1 Element equations: GM/WF
678(3)
11.8 Finite element formulations in R2 and R3
681(1)
11.9 Summary
681(2)
12 Convergence, Error Estimation, and Adaptivity
683(88)
12.1 Introduction
683(1)
12.2 h-, p-, k-versions of FEM and their convergence
684(5)
12.2.1 h-version of FEM and h-convergence
685(1)
12.2.2 p-version of FEM and p-convergence
686(1)
12.2.3 hp-version of FEM and hp-convergence
686(1)
12.2.4 k-version of FEM and k-convergence
687(2)
12.3 Convergence and convergence rate
689(4)
12.3.1 Convergence behavior of computations
690(2)
12.3.2 Convergence rates
692(1)
12.4 Error estimation and error computation
693(1)
12.5 A priori error estimation
694(25)
12.5.1 Galerkin method with weak form (GM/WF): self-adjoint operators
694(3)
12.5.2 GM/WF for non-self adjoint and non-linear operators
697(1)
12.5.3 Least-squares method based on residual functional: self-adjoint and non-self-adjoint operators
698(1)
12.5.4 Least-squares method based on residual functional for non-linear operators
699(3)
12.5.5 Integral forms based on other methods of approximation
702(1)
12.5.6 General remarks
702(1)
12.5.7 A priori error estimates: GM/WF and LSP
703(1)
12.5.7.1 Model problem 1: GM/WF
703(1)
12.5.7.2 Model problem 2: LSP
704(2)
12.5.7.3 Proposition and proof
706(6)
12.5.7.4 Proposition and proof
712(2)
12.5.7.5 Convergence rates
714(2)
12.5.7.6 Proposition and proof
716(3)
12.5.7.7 General Remarks
719(1)
12.6 Model problems
719(28)
12.6.1 Model problem 1: Self-adjoint operator, 1D diffusion equation
720(1)
12.6.1.1 GM/WF
721(6)
12.6.1.2 LSP, higher-order system (no auxiliary equation)
727(3)
12.6.2 Model problem 2: Non-self-adjoint operator, 1D convection-diffusion equation
730(1)
12.6.2.1 LSP: First order system
731(6)
12.6.2.2 GM/WF
737(1)
12.6.2.3 LSP: Higher order system (without auxiliary equation)
738(3)
12.6.3 Model problem 3: Non-linear operator, 1D Burgers equation
741(2)
12.6.3.1 LSP: Higher-order system (without auxiliary equation)
743(4)
12.6.3.2 GM/WF
747(1)
12.7 A posteriori error estimation and computation
747(4)
12.7.1 A posteriori error estimation
747(2)
12.7.2 A posteriori error computation
749(2)
12.8 Adaptive processes in finite element computations
751(16)
12.8.1 Adaptive processes for 1D convection-diffusion equation: non-self adjoint operator
752(1)
12.8.1.1 Adaptivity in the pre-asymptotic range: uniform h-refinement
752(1)
12.8.1.2 Adaptivity in the pre-asymptotic range: adaptive h-refinement
753(1)
12.8.1.3 Adaptivity in the pre-asymptotic range: graded h-rediscretizations
754(3)
12.8.1.4 General Remarks
757(1)
12.8.1.5 Adaptivity in the onset of asymptotic and asymptotic ranges: uniform p-refinement
758(1)
12.8.1.6 Adaptivity in the onset of asymptotic and asymptotic ranges: adaptive p-refinement
758(2)
12.8.1.7 Adaptivity in the onset of asymptotic and asymptotic ranges: adaptive h-refinement
760(1)
12.8.1.8 Adaptivity using higher geometric ratios for h-rediscretization at Pe = 1000 and Pe = 106
760(1)
12.8.2 Adaptive processes for 1D Burgers equation: non-linear operator
761(4)
12.8.3 Adaptive processes for 1D diffusion equation: self-adjoint operator
765(2)
12.8.4 General Remarks
767(1)
12.9 Summary
767(4)
Appendix A Numerical Integration using Gauss Quadrature
771(8)
A.1 Gauss quadrature in R1, R2 and R3
771(4)
A.1.1 Line integrals over Ωm = Ωε = [ --1, 1]
771(1)
A.1.2 Area integrals over Ωm = Ωε = [ --1, 1] × [ --1, 1]
772(1)
A.1.3 Volume Integrals over Ωm = Ωεησ = [ --1, 1] × [ --1, 1] × [ --1, 1]
773(2)
A.2 Gauss quadrature over triangular domains
775(4)
Index 779
Karan S. Surana attended undergraduate school at Birla Institute of Technology and Science (BITS), Pilani, India and received a B.E. in mechanical engineering in 1965. He then attended the University of Wisconsin, Madison where he obtained M.S. and Ph.D. in mechanical engineering in 1967 and 1970. He joined The University of Kansas, Department of Mechanical Engineering faculty where he is currently serving as Deane E. Ackers University Distinguished Professor of Mechanical Engineering. His areas of interest and expertise are computational mathematics, computational mechanics, and continuum mechanics. He is the author of over 350 research reports, conference papers, and journal papers.

J. N. Reddy is a Distinguished Professor, Regents Professor, and inaugural holder of the Oscar S. Wyatt Endowed Chair in Mechanical Engineering at Texas A&M University, College Station, Texas. Dr. Reddy earned a Ph.D. in Engineering Mechanics in 1974 from University of Alabama in Huntsville. He worked as a Post-Doctoral Fellow in Texas Institute for Computational Mechanics (TICOM) at the University of Texas at Austin, Research Scientist for Lockheed Missiles and Space Company, Huntsville, during l974-75, and taught at the University of Oklahoma from 1975 to 1980, Virginia Polytechnic Institute & State University from 1980 to 1992, and at Texas A&M University from 1992. Professor Reddy also played active roles in professional societies as the President of USACM, founding member of the General Council of IACM, Secretary of Fellows of AAM, member of the Board of Governors of SES, Chair of the Engineering Mechanics Executive Committee, among several others.
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