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| Prof. Carsten Könke |
| Institute of Structural Mechanics |
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| Course Format: |
| 4 hours of lectures (including exercises) per week; 15 weeks; 5 credits |
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| Course Contents: |
| Differential equations in strong and weak formulation |
| Principle of virtual work |
| Approximate solution techniques; solution errors |
| Formulation of element stiffness matrices |
| Isoparametric finite elements |
| Global stiffness matrix |
| Solution techniques for linear static problems |
| Mixed finite element models |
| Non-linear finite element analysis in solid mechanics
(geometrically and physically non-linear methods) |
| Solution of equilibrium equations |
| Error estimates and adaptive finite element methods |
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| Course Description: |
| Engineers working in design and maintenance-management operations today
have been used to the application of numerical simulation tools, in
order to predict various physical phenomena, e.g. in fluid flow or
structural mechanics problems. Different physical phenomena in nearly
all areas of applied engineering sciences can be transformed, by using
mechanical and mathematical abstraction techniques, into systems of
coupled differential equations. Assuming general geometry and boundary
conditions most of these equation systems can not be solved analytically. |
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| Using numerical discretization methods and applying modern computer
hardware opened a way to obtain an approximation of the exact solution
for these type of differential equations. Numerical discretization methods,
such as finite element procedures, have become an indispensable part of
engineering research as well as engineering analysis and design, for the
analysis of structures, solids, fluids and similar field problems.
The one-term course on finite element methods intends to offer
students an insight into the mechanical and mathematical foundations of
numerical discretization methods, especially finite element methods, as
well as demonstrating typical aspects of computer implementation of the
appropriate algorithms. |
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| The course is focused on the presentation of general principles in
computational differential equations, such as convergence and stability
of the approximate solution. Topics of error analysis and adaptive error
control will be addressed. Examples from structural and heat analysis
will present realization of numerical algorithms, based on finite element
procedures. An integral part of the course are exercise courses in the
computer lab, using mathematical manipulation tools, such as MAPLE and
MATHEMATICA as well as special teachware and general purpose finite
element programs. |
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| Structural response is often showing strong non-linear behavior, due to
geometrical and physical non-linear effects. Phenomena of large
displacements and changing boundary conditions in contact problems as
well as changing geometry due to damage and crack propagation can be
summarized as geometrical non-linear phenomena. Physical non-linear behavior
is describing all kinds of material non-linear effects, such as classical
plasticity. |
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| The extension of linear discretization methods, such as linear finite
element methods, into the non-linear regime leads to a variety of new
problems, starting from continuum mechanics formulation and ending with
solution techniques for sets of non-linear differential equations. |
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| The second part of the course, therefore, intends
to offer students an insight into the mechanical and mathematical
foundations of non-linear numerical discretization methods, especially
finite element methods for geometrical and physical non-linear problems,
as well as demonstrating typical aspects of computer implementation of
the appropriate algorithms. Thereby, a focus is on the presentation of general principles in
computational differential equations for non-linear problems, such as
convergence and sensitivity of the approximate solution. Topics of
non-linear discretization techniques for full geometrical non-linear
analysis, stability analysis and stiffness softening problems will
be addressed. |
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| Recommended Reading: |
| K.-J. Bathe (1996). "Finite Element Procedures." Prentice Hall, Englewood Cliffs.
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