CE 324      Mathematical Concepts and Applications in Civil Engineering                       3-1-0-8

 

Pre-Requisite: MA 101, MA 102 and MA 201

Syllabus: Special matrices: stiffness, circulant and Toeplitz matrices, with physical connections to civil and mechanical systems (spring mass systems) with different boundary conditions; Linear Algebra refresher; Vectors, matrices, vector-spaces and subspaces. Introduction to row-space, column-space and null-space; Solving Linear equations: existence and uniqueness of solutions. Elimination, interpretation of A\I, interpretation of I as an unit impulse matrix, LU factorization, LDL factorization, Cholesky factorization; Eigenvalues, eigenvectors and EVP: Properties of eigenvectors, matrix diagonalization, application to vector differential equations, applications to structural dynamics problems, eigenvalues and stability, applications to multivariate data analysis, Principal Component Analysis and model order reduction, Matlab command: eigshow; Numerical Linear Algebra: Power and Rayleigh quotient iterations, Gram Schmidt orthogonalization, LU, QR and SVD, condition numbers and norms; Delta and Green’s functions: Conceptualization of concentrated load; properties of Dirac delta function, derivatives and integrals of Delta function, Macaulay brackets and applications to beam deflections, introduction to Green’s functions. Introduction to Differential and Difference equations: Idea of forward, backward and central differences and Taylor’s series, finite difference equations, boundary conditions, accuracy and convergence, boundary value problems (BVPs), application to Euler column buckling, MATLAB examples; Finite Differences in Time: Numerical solutions of ordinary differential equations (IVPs), methods based on Taylor’s series, Runge Kutta methods, forced vibration of linear and nonlinear dynamic systems, MATLAB ode-45, ode-123s commands; Least Squares: overdetermined and underdetermined set of equations, linear and nonlinear regression, underdetermined equations and sparsity, least squares by calculus and linear algebra, computational and recursive least squares; Introduction to Function Approximation: Introduction to function spaces, function approximation, Fourier series, interpolation functions, continuous least squares, collocation method, weighted residuals, Galerkin method, introduction to finite elements in civil engineering, equivalence between force equilibrium (strong form) and energy formulation (weak form), application examples on bar extension, beam bending, 2D plane stress and strain problems; Introduction to Poisson’s and Laplace’s Equations: Application to Torsion of Circular and Rectangular shafts, flow through porous media, Fast Poisson solvers.

Texts:

  1. Strang, Computational Science and Engineering, Wellesley-Cambridge Press, 2007
  2. Strang, Linear Algebra and its Applications, 4th edition, Cengage Learning, 2005
  3. G. Kelly, Advanced Engineering Mathematics with Modeling Applications, 2008, CRC.
  4. Kreyszig, H. Kreyszig and E. Norminton, Advanced Engineering Mathematics, 10th edition, John Wiley & Sons, 2011

References:

  1. B. Magrab, S. Azarm, B. Balachandran, J. Duncan, K. Herold and G. Walsh, An Engineer’s Guide to MATLAB, 3rd edition, Pearson, 2010
  2. Holzbecher, Environmental Modelling using MATLAB, 2nd edition, Springer, 2012.
  3. S. Desai and J. T. Christian, Numerical Methods in Geotechnical Engineering, McGraw-Hill, 1977.

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