## 주메뉴

### Basics of Computational Science and Engineering

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자연과학 >수학ㆍ물리ㆍ천문ㆍ지리 >수학
• 등록일자
2010.10.01
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5,779
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This course is intended for graduate students (or undergraduate) who need a rapid and uncomplicated introductions to the field of applied mathematics involving computational linear algebra and differential equations. The lecture has two themes-how to understand equations, and how to solve them . This course include numerical linear algebra(QR,SVD, singular system), Newton’s method for minimization, Equilibrium and stiffness matrix, Least squares, Nonlinear problems, Covariances and Recursive Least squares, Differential equations and finite elements, Finite Difference and Fast Poisson, Boundary value problems in Elasticity and Solid mechanics
Graph models and Kirchhoffs laws 1

#### 차시별 강의

 1 Graph models and Kirchhoffs laws 1 The incidence matrix Kirchhoffs current law (KCL)Kirchhoffs voltage law (KVL) 2 Graph models and Kirchhoffs laws 2 The incidence matrix Kirchhoffs current law (KCL)Kirchhoffs voltage law (KVL) 3 Graph models and Kirchhoffs law / Newtworks and transfer functions Assembling the matrix K = A^TCAThe saddle point KKT matix Loop equation fow W and impedance Z 4 Newtworks and transfer functions Time domain versus frequency domain Laplace transform 5 Newtworks and transfer functions / Nonlinear problem Underdamping and overdampingNewtons method 6 Nonlinear problem / Differential equations and finite elements Newtons method Minimizing P(u) Steepest decentSecond - order equations 7 Differential equations and finite elements (1) The A^TCA framework for a hanging barThe transpose of A = d/dxGalerkins method(FEM) 8 Differential equations and finite elements (2) Construction of the Finite Element MethodLinear Finite element Comparison with finite differences 9 Differential equations and finite elements / Cubic splines and fourth-order equations More accurate finite elements Fourth order equations : Beam bending 10 Cubic splines and fourth-order equations Cubic splines for interpolation Continuity conditions Cubic finite elements 11 Gradient and divergence The divergence The Divergence theorem.29분 30초 이후부터 음성이 나오지 않습니다 12 Gradient and divergence Definition of divergence and curl. Stokes and Greens TheoremThe Gauss - Green formulaGradiant and divergence plane vertor field v, w 13 Gradient and divergence Gradients and irrotational velocity fieldContituous form of Kirchhoffs voltage lawPlane gradient fields 14 Gradient and divergence / Laplaces equation Plane gradient fieldsSolutions of Laplaces equation The Cauchy-Riemann equations 15 Laplaces equation Polar coordinates :laplaces equation in a circle 16 Laplaces equation / Finite differences and fast poisson solvers · Solver using eigenvalues · Fast Poisson solvers · Trial and test functions : Galerkins method 17 Finite differences and fast poisson solvers / The finite element method · Solver using eigenvalues · Fast Poisson solvers · Trial and test functions : Galerkins method 18 The finite element method · Paramid functions · Element matrices and element vectors 19 The finite element method · Element matrices and element vectors · Boundary conditions come last 20 The finite element method · Element matrices in two dimentions · Quadrilateral elements 21 Introduction of this course / Four special matrices Introduction of this courseProperties of four special of matrices : K, C, T, B 22 Four special matrices / Differences, dirivatives, boundary conditions Properties of four special of matrices : K, C, T, B 23 Differences, dirivatives, boundary conditions / Elimination leads to k = LDL^T Properties of four special of matrices : K,C,T,BGaussian elimination and.LU decomposition. 24 Elimination leads to k = LDL^T Sysmmetry convert K = LU to K = LDL^TThe determinant of Kn 25 Inverse and dalta functions Delta functions and Greens function 26 Inverse and dalta functions / Eigenvalaue and Eigenvectors Discrete vectors: Load and Step and RampGreens function and the inverse matrixDiagonalizing a Matrix 27 Eigenvalaue and Eigenvectors Ax = λx and A^k x = λ^k x and diagonalizing A 28 Eigenvalaue and Eigenvectors The power of matrixApplication to vector differential equationsEigenvectors and derivatives and differences 29 Eigenvalaue and Eigenvectors Eigenvectors of Kn :Discrete SinesEigenvectors of Bn :Discrete Cosines 30 Eigenvalaue and Eigenvectors / Positive Definite Matrices Eigenvectors of CnThe Fourier MatrixEnergy or quadratic form 31 Positive Definite Matrices (1) Minimum Problem in n dimensionsNewtons Method for Minimization 32 Positive Definite Matrices (2) Minimum Problem in n dimensions Newtons Method for Minimization 33 Numerical Linear Algebra: LU, QR, SVD 1 Orthogonal MatricesOrthogonalization A = QR Singular Value Decomposition 34 Numerical Linear Algebra: LU, QR, SVD 2 Singular value DecompositionThe Pseudoinverse Condition numbers 35 Numerical Linear Algebra: LU, QR, SVD 3 Condition numbers and Norms 36 Equilibrium and the Stiffness Matrix Stiffness matrix and solution Minimum principle 37 Oscillation by Newtons Law (1) One mass and One SpringMotion around a circleLine of masses and spring 38 Oscillation by Newtons Law (2) M - orthogonality Total energy is conserved Applied force and resonance 39 Least squares for rectangular matrices Least squares by calculusLeast squares by linear algebra Computational Least squares

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