Solution Methods 1. Why use the Penalty Method for implementing boundary conditi
ID: 1859938 • Letter: S
Question
Solution Methods
1. Why use the Penalty Method for implementing boundary conditions
2. What are the advantages/disadvantages of direct versus iterative solutions?
3. What is the difference between implicit and explicit iterative solutions?
4. Be able to provide example methods for each technique
Dynamic Analysis
1. Know how to derive the FE equations for a dynamic system.
2. Know how to solve the FE equations for a dynamic system.
Beams and Shells
1. How are beam element formulations different from continuum mechanics formulations (i.e.,
plane stress elements)?
2. What does %u201Cplane sections remain plane%u201D mean and how does it apply to beam elements?
3. How do we account for both bending and axial forces in beams?
4. What are (is) the main difference(s) between plate and shell elements?
5. What type of approximation functions are used in Beam Element formulations and how do
they differ from Lagrangian Approximation functions?
Explanation / Answer
Solution Methods
1) A penalty method replaces a constrained optimization problem by a series of unconstrained problems whose solutions ideally converge to the solution of the original constrained problem. The unconstrained problems are formed by adding a term to the objective function that consists of a penalty parameter and a measure of violation of the constraints. The measure of violation is nonzero when the constraints are violated and is zero in the region where constraints are not violated.
Hence it is easy to use than the lagrangian multiplier and other approaches in certain problems.
2)Advantage of direct solution lies in the fact that we can go on solving a problem in an analytical method and definitely arrive at an answer and have a better understanding of the problem and the underlying concepts. This is not the case in iterative method.
Disadvantage of this method is that we might have to solve higher order differential equations which might not be easily solvable and it is time consuming. But iterative method allows us to reach a solution at a much faster pace, and hence is preffered at times over a direct solution.
3)Implicit iteration allows entails certain methods that are tested and expects us to follow it without much logic and gives us the solution. Explicit iteration on the other hand helps us understand why we are doing the iteration in that certain way.
Beams and Shells
1) Beam element formulation breaks the beam down into small elements and analyses each aspect of forces acting on it to give us the final formulation required. But in mechanics we consider the beam to be a total element and formulate the analysis based on the assumption that it is a single entity.
2)Plane section remains same means that the section of plane in a beam during analysis is considered to be undeformed during analysis of the beam.
In case of beams we usually take the cross section of beams to be a plane section while the lenght of the beam may undergo deformation due to loading.
3)We account for both bending and axial forces in beams by choosing appropiate co ordinate system and analysing forces by representing them as vectors. This helps us to analyze forces in all direction which gives us a comprehensive answer consisting of bending, axial forces as well as moments and torques applied to the beam.
4)Main difference between plate and shell element is their shape and hence the different methods of analysis of both the element.
5) Many approximations may be used in beam element formulation like the simpsons third rule, or that we take the beam element to be the single unit and that stiffness k of the beam remains unchanged due to loading.
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