The most common objective in topology optimization is to minimize compliance. The compliance is depending on the global stiffness matrix and displacement field.
How could we find the global stiffness matrix?
Finding it using hand calculations is not feasible all the time!
Note: we need to define the stiffness matrix as a function of area (pseudo density).
Hello,
That is an interesting topic, with which I'm not acquainted, however after a quick read to Wikipedias webpage (https://en.wikipedia.org/wiki/Topology_optimization). I would guess it can be accomplished in many ways, but I'm thinking about an iterative solution in which the elements with the lowest stress values would disappear or reduce their size. From a FEM background: Global stiffness matrix K_G can be calculated by assembling the individual stiffness matrices K_E, which are calculated from the integral over element volume of Transpose(B) *E *B dV in which B is the matrix of the derivatives of u field [D{u}]. The displacement field can be stated in terms of the shape functions and the nodal displacement.
Maybe is not what you are looking for but I hope it helps
You might be interested in the free education edition of PERMAS. See https://www.intes.de/edu and the PERMAS4EDU project on Researchgate. Topology optimization is integrated in the FEA solver.
You can easily perform topology optimization with ANSYS finite element software. For example, how about starting with this study: Article Topological Optimization of Continuum Structure based on ANSYS
And these explanations: https://www.ansys.com/blog/topology-optimization-motorcycle
And with this video: https://www.youtube.com/watch?v=mAwtK8ITSfk
Thank you Juan SEBASTIAN Leon Becerra
In that case, is there any way to find the shape functions for the whole structure without using hand calculations?
Thank you Nils Wagner and @ Ferhat Çeçen
Actually, I have shown my interest in INTES Engineering Software and applied for a license for educational purposes but still have not received any response regarding that!
The reason that I am not using the integrated TO in FEA packages (like TOSCA, ANSYS, and others) is that I am working on the basic principles and algorithms for research purposes, not industrial ones.
Ahmed Mohamed Jubartalla Ali Happy to know that you have received the needed information. Good luck!
Ahmed Mohamed Jubartalla Ali, you can use isoparametric elements with the coordinate transform functions. If you want a computational implementation, books from Carlos Felipa are useful: https://vulcanhammernet.files.wordpress.com/2017/01/ifem.pdf I have the complete series (4 books Introductory FEM, Nonlinear Fem, Advanced Fem, and AVVM). Best of lucks!
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