I have the two configurations of a piece of soft tissue (Undeformed and deformed states) modeled in ABAQUS. I already have the initial and final coordinates of all nodes (over one hundred thousand nodes) corresponding to the undeformed and deformed configurations, respectively. I am trying to build a mapping function capable of deforming the undeformed shape and registered it onto the deformed one based on the initial and final coordinates obtained by the finite element procedure. In other words, is there any way to find a function or whatever else to be applied to a given geometry and gives you the desired configuration?
I truly appreciate your recommendations, in advance.
I would like to suggest you some related basics : https://www.sciencedirect.com/topics/engineering/initial-undeformed-configuration and https://mae.ufl.edu/nkim/egm6352/Chap3.pdf
Dear Shahrokh Rahmani
The idea:
Let f be a mapping from the set of points {Ai} into the set of deformed points {Bi}.
Our goal is to find a mapping f with (n) branches {fi} up to the dimension of the given data.
Method description:
Assume that f₁(A₁A₂) = B₁B₂
The image of the line segment A₁A₂ is a line segment B₁B₂.
So we seek f₁((1-t)A₁ + tA₂) = (1-s)B₁+sB₂ where 0 ≤ s,t ≤1.
Assume that f₁(x,y,z)=(ax+b, cy+d, ez+f)
Solve f₁(A₁)= B₁ and f₁(A₂)=B₂,
it represents six equations in 6 unknowns a,b,c,d,e and f.
Therefore the diffeomorphism f₁ is entirely determined,
that maps the segment A₁A₂ onto B₁B₂.
We repeat the same procedure to determine any branch fi.
My advice is to design a simple program to do the job.
Best regards
Thank you all for your time and useful suggestions. Muhammad for the paper, Issam and Peter for the solution method. Let me explain more. Actually, I have a piece of meat in a curved configuration, i.e., it is lying on a curved substrate. My aim, initially, is to track its configuration and finds its shape when it is put on a flat substrate. For this, I constructed the solid shape of the meat using CT scan image reconstruction and then imported into ABAQUS for deformation analysis using finite element method. I mean, I applied body force to the curved meat in order to reach the flat configuration. I have done this and compared the final configuration of the meat predicted by finite element with that in reality on a flat substrate. I have done this procedure for two different pieces of meat (approximately the same size and the same configuration on the same curved substrate). Now, I have the undeformed and deformed configurations of the two pieces and their corresponding nodes’ coordinates. The reason why I am looking for such formula (mapping function) is to find an applicable tool (method or operator function) to be used to deform the meat from the curved configuration in order to approximate its flat state. Since we do not want to use finite element method to reach this and we actually aim at finding a way to predict the flat configuration for many cases using a generalized mapping function rather than a finite element method. The problem is that the number of mesh elements and nodes are different for each individual case. We actually want to find a generalized function based on the results that we got from the finite element method to be applied to a curved meat and gives a flat configuration, automatically.
Best regards,
Shahrokh
Dear Shahrokh Rahmani
I think you can use the previously proposed method where the coordinates of
Ai and Bi are variables, and fi can be replaced by a matrix Mi that transforms the vector AiAj into the vector BiBj. Then apply a suitable simple program that can transform the initial region into the new one via the family { Mi }.
Where Mi is completely determined by the initial inputs.
Best regards
Dear Issam,
Thank you so much, I agree with you and I am working on a simple model with a coarse mesh (for a start) to see the outcome
Best regards,
Shahrokh.
Dear Peter,
Thank you very much for your suggestion. Honestly, I still have problems with this issue. You are right. I am trying to find a simple formula or a mapping function relating final shape to initial shape. Actually, the model is a belly tissue, which is not uniform in thickness, and when it deforms the thickness will change slightly in some regions. As you mentioned, it is actually incompressible and the volume does not change during the deformation. However, as I mentioned I want to find this mapping function (or a simple formula) in such a way if I apply this function to any other bellies in the curved position (lying on the same curved substrate), it gives me, roughly, the final configuration on a flat substrate. It is worth mentioning that different bellies are a bit different from each other in shapes and size and the number of nodes from mesh generation will vary across different bellies. So, I only want to do this on one belly (at most two cases; because I can find the final shape from initial shape by finite element method) and find a generalized function capable of doing this deformation for more bellies.
Best regards,
Shahrokh
Dear Peter,
Thank you for your reply. I am currently working in the field of solid mechanics particularly on the soft tissue deformation. I have also done my PhD on fluid-structure interaction in cardiovascular modeling. I have also some experience working with CT and MRI images in terms of image segmentation and visualization. I have not gone that much through mathematical modeling, functional mapping and shape analysis. This is the first time that I am looking for such mapping function for a non-uniform and complex geometry.