PENG9570 Applied Mathematical Modelling and Analysis Course description

All support material is allowed.

A thorough knowledge of advanced calculus, including ordinary and partial differential equations. It is a great advantage if students are familiar with linear algebra and Fourier and Laplace transform theory. In terms of programming, some experience in implementing various numerical methods, including schemes for solving partial differential equations is recommended. Some knowledge of mathematical analysis, modern physics or physiology is recommended, depending on their specialisation.

Pass or fail.

Two internal examiners. External examiner is used periodically.

Introductory module:

• Principles of modelling and derivation of mathematical models
• Analysis of ordinary differential equations (ODEs)
• Linear partial differential equations (PDEs)
• Prominent results from functional analysis and their application to ODEs and PDEs
• Numerical methods for computing of solutions of PDEs

Functional analysis:

• Completeness for normed spaces
• Hilbert spaces, compact and diagonalisable operators
• Theory of topological vector spaces
• Test functions, distributions and the Fourier transform
• Sobolev spaces and fundamental solutions of partial differential equations

Biosystems:

• Mathematical models for biological systems
• Analytical and numerical methods for simulation of system response
• Actuators and sensors for stimulation and measurements of biological systems
• Interaction of biological and measurement system

Modern physics:

• Monte Carlo techniques
• Splines and other expansion techniques
• Applications of expansions in spherical harmonics
• Numerical problems in general relativity and quantum physics
• Manifolds with geometric structures central to physics and engineering.

Within all specializations, the content may be adjusted to accommodate for the research area of each PhD candidate.

The course is based on knowledge and skills within solid mechanics, statics, design of reinforced concrete and steel structures as well knowledge in finite element method (FEM) in structural analysis and design.

Introduction to the theoretical foundation for nonlinear finite element analysis of civil engineering structures. Classification of nonlinearities (geometrical, material and boundary conditions). Strain and stress measures for large displacements/deformations. Mathematical models for elastic and elastoplastic materials like reinforced concrete (RC), steel,etc. Geometrical stiffness and linearised buckling. Numerical integration of dynamically excited systems. Implicit/explicit time integration. Incremental-iterative solution methods for nonlinear static and dynamic problems. Modelling of nonlinear boundary conditions. The course will also expose students to recent trends and research areas in FEM.

An individual, oral examination. The examination will address both general topics from within the course and the specific project developed by the student.

The oral examination cannot be appealed.

The student's own project.

Pass or fail.

Two examiners. External examiner is used periodically.