SYLLABUS:-
UNIT-I
Boundary Value Problems and the Need for Numerical Discretisation: Introduction, examples of Continuum problems, history of finite element method.
Weighted residual methods: Approximation by trial functions, weighted residual forms, piecewise trial functions, weak formulation, Galerkin method, examples of One-, two- and three -dimensional problems.
UNIT-II
Higher order finite element approximation: Degree of polynomial in trial functions and rate of convergence, the patch test, shape functions for C0 and C1 continuity, one-, two-and three-dimensional shape functions.
Isoperimetric formulation: The concept of mapping, isoperimetric formulation, numerical integration, mapping and its use in mesh generation.
UNIT-III
Variational Methods: Variational principles, establishment of natural Variational principles, approximate solution of differential equations by Rayleigh-Ritz method, the use of Lagrange multipliers, general Variational principles, penalty functions, least-square method.
Partial discretisation and time-dependent problems: Partial discretisation applied to boundary value problems, time-dependent problems via partial discretisation, analytical solution procedures, finite element solution procedures in time domain.
UNIT-IV
Generalized finite elements and error estimates: The generalized finite element method, the discretisation error in a numerical solution, measure of discretisation error, estimate of discretisation error.
Coordinate Transformation: Transformation of vectors and tensors, transformation of stiffness matrices, degree of freedom within elements, condensation, condensation and recovery algorithm, sub structuring, structural symmetry. [
UNIT-I
Boundary Value Problems and the Need for Numerical Discretisation: Introduction, examples of Continuum problems, history of finite element method.
Weighted residual methods: Approximation by trial functions, weighted residual forms, piecewise trial functions, weak formulation, Galerkin method, examples of One-, two- and three -dimensional problems.
UNIT-II
Higher order finite element approximation: Degree of polynomial in trial functions and rate of convergence, the patch test, shape functions for C0 and C1 continuity, one-, two-and three-dimensional shape functions.
Isoperimetric formulation: The concept of mapping, isoperimetric formulation, numerical integration, mapping and its use in mesh generation.
UNIT-III
Variational Methods: Variational principles, establishment of natural Variational principles, approximate solution of differential equations by Rayleigh-Ritz method, the use of Lagrange multipliers, general Variational principles, penalty functions, least-square method.
Partial discretisation and time-dependent problems: Partial discretisation applied to boundary value problems, time-dependent problems via partial discretisation, analytical solution procedures, finite element solution procedures in time domain.
UNIT-IV
Generalized finite elements and error estimates: The generalized finite element method, the discretisation error in a numerical solution, measure of discretisation error, estimate of discretisation error.
Coordinate Transformation: Transformation of vectors and tensors, transformation of stiffness matrices, degree of freedom within elements, condensation, condensation and recovery algorithm, sub structuring, structural symmetry. [
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