This course has four parts. The first part is concerned with variational method, Ritz's method and weighted residual method, which are closely related to the most popular numerical method, i.e., finite element method (FEM). In the second and third parts, theories and numerical schemes of FEM and boundary element method (BEM) are explained, respectively. In the fourth part, theory and application of inverse analyses are introduced for some typical examples in civil engineering problems.
Problems related in civil engineering are mainly divided into forward problems and inverse problems. Forward problems are well-posed ones, for which existence and uniqueness of solutions are guaranteed. To solve forward problems, many numerical approaches such as finite difference method, FEM and BEM have been proposed. On the other hand, inverse problems have generally no guarantee for uniqueness and existence of solutions. However, since many important engineering problems require inverse analyses, there exist various methodologies to solve inverse problems. The first three parts of this course are related to the forward problems, while the last part is to inverse ones.
By the end of this course, students will be able to:
1. Explain the relationship among variational method, Ritz's method, weighted residual method and finite element method (FEM).
2. Explain theories and numerical schemes of FEM and boundary element method (BEM).
3. Make numerical codes of FEM and BEM for simple problems.
4. Explain theory and application of inverse analysis.
forward problem, inverse problem, variational calculus, weighted residual method, numerical analysis, approximation, finite element method, boundary element method, singular value decomposition
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
This course will be mainly provided in lecture style. However, practical exercises with programming for finite element method and boundary element method are included to allow students to get a good understanding of fundamentals for numerical analysis.
|Course schedule||Required learning|
|Class 1||Variational method - one dimensional problems||Explain a variational method for one dimensional problem and solve one dimensional variational problem.|
|Class 2||Variational method - multidimensional problems||Explain a variational method for multidimensional problem and solve multidimensional variational problem.|
|Class 3||Variational method with constraint conditions||Explain variational method with constraint conditions, and solve related variational problems.|
|Class 4||Ritz’s method||Explain Ritz’s method, and solve related problems.|
|Class 5||Weighted residual methods||Explain weighted residual methods, and solve related problems.|
|Class 6||Finite element method - one dimensional problems||Explain the formulation of finite element method for one dimensional problem.|
|Class 7||Finite element method - multidimensional problems||Explain the formulation of finite element method for multidimensional problem.|
|Class 8||Finite element method - numerical schemes & examples||Explain numerical schemes of finite element method, and show examples.|
|Class 9||Boundary element method - one dimensional problems||Explain the formulation of boundary element method for one dimensional problem.|
|Class 10||Boundary element method - multidimensional problems||Explain the formulation of boundary element method for multidimensional problem.|
|Class 11||Boundary element method – numerical schemes & examples||Explain numerical schemes of boundary element method, and show examples.|
|Class 12||Numerical programming||Perform numerical programming for finite element method or boundary element method.|
|Class 13||Inverse problem - introduction||Explain the concept of inverse problem.|
|Class 14||Inverse problem - math fundamentals and formulation||Explain math fundamentals and formulation of inverse problem.|
|Class 15||Inverse problem - examples||Explain examples of inverse problem in engineering fields.|
Lecture materials will be uploaded on OCW-i.
Assignment (40%) and examination (60%)
Students must have successfully completed Computers and Fundamental Programming (CVE.M301) and Computers and Applied Programming (CVE.M302), or have equivalent knowledge.