Solving Elasticity Problems with Finite Element Method

In summary, the conversation discusses the challenge of incorporating boundary conditions in a program that solves elasticity problems with the finite element method. It is suggested to use constant displacement boundary conditions by replacing variables and dropping corresponding equations. However, if the boundary conditions involve pressures instead of displacements, it is recommended to replace them with equivalent forces on nodes. This would generate work terms and the nodes would collectively minimize the sum of work terms and strain energy to find the solution. Another approach is to leave the unknown pressures as variables in the Lagrange equations and obtain a linear system of equations to solve for both unknown displacements and pressures.
  • #1
Lojzek
249
1
I am trying to make a program that solves elasticity problems with finite element method and
I don't understand how to bring in boundary conditions.

Constant displacement boundary conditions seem simple: replace variables that represent the displacements at surface nodes with the prescribed constants and drop corresponding Euler-Lagrange equations for this variables.

But what if boundary conditions define pressures on the boundaries instead of displacements?
And how do we deal with the problem with both types of boundary conditions?
 
Engineering news on Phys.org
  • #2
How about replacing pressures on boundaries with equivalent forces on nodes? Then when a node moves, a work term is generated. The nodes will collectively displace to minimize the sum of work terms and strain energy in the body.

(I haven't tried this personally, but it may give you some ideas.)
 
  • #3
I think I got the solution now. The unknown pressures on element surfaces should be left as unknown variables in the Lagrange equations together with unknown displacements and a sistem of linear equations can be obtained, where the unknown vector contains both unknown displacements and pressures.
Replacing pressures on boundaries with equivivalent forces on nodes would probably work in a similar way. Then unknown displacements and unknown forces would be determined by the linear system.
 

1. What is the Finite Element Method (FEM)?

The Finite Element Method is a numerical technique used to solve complex engineering and scientific problems. It involves dividing a continuous system into smaller, simpler elements, and using mathematical equations to model the behavior of each element. By combining the behavior of all the elements, the overall behavior of the system can be accurately predicted.

2. How does the FEM work for solving elasticity problems?

The FEM works by creating a mesh of the system, where each element is represented by a set of nodes and connected by a series of mathematical equations. These equations are then solved to determine the displacement, strain, and stress of each element. The overall displacement and stress of the system can then be obtained by combining the results from all the elements.

3. What are the benefits of using FEM for elasticity problems?

The FEM offers several advantages for solving elasticity problems. It allows for complex geometries and boundary conditions to be easily modeled, and it can handle a wide range of materials and loading conditions. It also provides accurate results with relatively few computational resources, making it a cost-effective solution for many engineering applications.

4. Are there any limitations to using FEM for elasticity problems?

While FEM is a powerful tool, it does have some limitations. It requires a significant amount of pre-processing to create the mesh and define the equations for each element. It also relies on assumptions and simplifications, so the accuracy of the results may be affected by the quality of the mesh and the chosen element type. Additionally, FEM is not suitable for problems involving large deformations or failure of materials.

5. What are some real-world applications of FEM for elasticity problems?

FEM has a wide range of applications in various fields, including mechanical, civil, and aerospace engineering, as well as materials science and biomechanics. It can be used to analyze structures such as bridges and buildings, simulate the behavior of materials under different loading conditions, and optimize the design of mechanical components. FEM is also commonly used in research and development to study the behavior of new materials and structures.

Similar threads

Replies
3
Views
756
Replies
4
Views
706
Replies
6
Views
2K
  • Mechanical Engineering
Replies
9
Views
1K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
4
Views
1K
Replies
1
Views
1K
  • Mechanical Engineering
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
4
Views
1K
Replies
4
Views
809
  • Engineering and Comp Sci Homework Help
Replies
7
Views
621
Back
Top