Numerical solving of elastomechanic problems

[Lojzek]

I want to make a simple program for numerical solving of elastomechanic problems. The basic task it should perform is the calculation of deformation of a two dimensional homogeneous, isotropic body in the field of gravity and/or deformation under external force.
The basic idea for each time step is:

1. define the displacement in a finite number of points
2. calculate the strain tensor using finite diferences of displacement
3. calculate stress tensor (Hooke's law)
4. use stress tensor and second Newton's law to predict acceleration of each point
5. increase the speed and displacement at each point

The boundary condition should be either a fixed displacement or stress tensor.
Alternative method would be minimization of elastic energy (however this would only give the stationary state, not a movement towards it).

Questions:

Do I need to calculate derivatives of displacement with second order accuracy? (I seems so, since the acceleration depends on the changes of strain tenzor) If this is the case, how should I achieve second order accuracy near the boundary?
I planned to use central diferences (second order accuracy) and a square mesh, but it seems that on the boundaries one would have to know the displacement two steps away from the last calculated point: (but there is only one boundary condition).

Please let me know if you know a good book or internet site about numerical methods for solving this type of problems.

 

[Jhero]

Thank you for your interest in developing a numerical program for solving elastomechanic problems. It is a complex and challenging task, but with the right approach and methods, it can be achieved.

To answer your first question, yes, it is necessary to calculate the derivatives of displacement with second order accuracy. This is because the acceleration depends on the changes in the strain tensor, which in turn depends on the changes in displacement. In order to achieve second order accuracy near the boundary, you can use a technique called "ghost points." This involves creating additional points outside of the boundary, which are used to calculate the derivatives near the boundary. These ghost points can be assigned values based on the boundary conditions.

Using central differences and a square mesh is a good approach, but as you mentioned, it may be challenging to implement near the boundaries. Another alternative is to use a non-uniform or adaptive mesh, which can better capture the behavior near the boundaries. Additionally, you can also consider using higher order finite difference schemes, such as the fourth-order accurate compact schemes, which can provide even better accuracy near the boundaries.

As for resources, there are many books and websites available that discuss numerical methods for solving elastomechanic problems. Some recommended books include "Numerical Methods in Engineering with Python" by Jaan Kiusalaas, "Numerical Methods for Engineers" by Steven Chapra and Raymond Canale, and "Numerical Methods for Scientists and Engineers" by Richard Hamming. You can also find many online resources and tutorials on websites such as NAFEMS (www.nafems.org) and COMSOL (www.comsol.com).

I hope this helps and wish you success in developing your program.