Basics of Computational Science and Engineering

연세대학교
정윤모

페이스북 트위터 카카오톡 네이버밴드 링크복사

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자연과학 >수학ㆍ물리ㆍ천문ㆍ지리 >수학
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2010.10.01

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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

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※ 수강확인증 발급을 위해서는 수강신청이 필요합니다

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^TCA The 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 overdamping Newtons method
6.	Nonlinear problem / Differential equations and finite elements	Newtons method Minimizing P(u) Steepest decent Second - order equations
7.	Differential equations and finite elements (1)	The A^TCA framework for a hanging bar The transpose of A = d/dx Galerkins method(FEM)
8.	Differential equations and finite elements (2)	Construction of the Finite Element Method Linear 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 Theorem The Gauss - Green formula Gradiant and divergence plane vertor field v, w
13.	Gradient and divergence	Gradients and irrotational velocity field Contituous form of Kirchhoffs voltage law Plane gradient fields
14.	Gradient and divergence / Laplaces equation	Plane gradient fields Solutions 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 course Properties 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,B Gaussian elimination and.LU decomposition.
24.	Elimination leads to k = LDL^T	Sysmmetry convert K = LU to K = LDL^T The 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 Ramp Greens function and the inverse matrix Diagonalizing 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 matrix Application to vector differential equations Eigenvectors and derivatives and differences
29.	Eigenvalaue and Eigenvectors	Eigenvectors of Kn :Discrete Sines Eigenvectors of Bn :Discrete Cosines
30.	Eigenvalaue and Eigenvectors / Positive Definite Matrices	Eigenvectors of Cn The Fourier Matrix Energy or quadratic form
31.	Positive Definite Matrices (1)	Minimum Problem in n dimensions Newtons 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 Matrices Orthogonalization A = QR Singular Value Decomposition
34.	Numerical Linear Algebra: LU, QR, SVD 2	Singular value Decomposition The 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 Spring Motion around a circle Line 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 calculus Least squares by linear algebra Computational Least squares