In 99-line topology optimization code of Prof. O.Sigmund the FEM is done without any gauss-integration carried out? I was wondering if someone could tell me the underlying theory behind this computationally efficient code. The link to the paper and the code is as follows.
http://www.topopt.dtu.dk/?q=node/2
Yes, you can do FEM without doing any Gauss integration.
You can integrate your functions analytically beforehand and use that expressions in your code. That works fine in the case of elements without distortion, since the Jacobian is the identity matrix. For the general case is difficult to find those expressions or they can be found but they are so extensive that it's not worth to use in your code. Particularly, in the 99-line code they already know the form of the matrices because of analytic integration.
Yes, you can do FEM without doing any Gauss integration.
You can integrate your functions analytically beforehand and use that expressions in your code. That works fine in the case of elements without distortion, since the Jacobian is the identity matrix. For the general case is difficult to find those expressions or they can be found but they are so extensive that it's not worth to use in your code. Particularly, in the 99-line code they already know the form of the matrices because of analytic integration.
Thank you for your answer Zapata. Could you please give reference to some article so that I can understand its theory more clearly.
Regards,
I don't remember seeing a paper like that. I suggest you to check the chapters 3 and 4 of this reference:
- Bathe, Klaus-Jürgen. Finite element procedures. Klaus-Jurgen Bathe, 2006.
I also attached the derivation done in Maxima (a computer algebra system) and a pdf with the equations, the notation is the same as used in Bathe's book.
If you understand the nature of the method, you would get that... Maybe a good option is to study a variational method, like Ritz-Galerkin method.
I am agree with Zapata. If the Gauss integration is used, the incompressible material can not be optimized, because the Jacobian is zero as the Poisson's ratio is 0.5. However, O.Sigmund's can successfully solve the incompressible material design problem. So, I think the Gauss Integration is not been utilized in 99-line topology optimization method.
Dear Waseem, there are FEM methods without using "Gauss" integration. However, in the source code of the paper "A 99 line...", there is integration step, the integration is considered with constant function in local element (such as constant strain element, or you can say only the center Gauss point is considered), hence only the volume of local element need to be considered. In the source code, parameter "volfrac" represents the volume of the local element.
Deat Yiming, the code in mention does not use constant elements. They use four-nodes isoparametric elements. If you read from lines 91 to 100 you will see this, the same matrices I showed before.
Nevertheless, it is true that you don't need Gauss integration for three-nodes elements since they are linear, they can even be deformed and the integrals are computed analytically.
http://www.topopt.dtu.dk/files/top.m
Dear Nicolas, I agree, it is four-nodes isoparametric elements. But you know actually assuming constant strain in this type of element (even higher order) is also very common, this is called reduced integration strategy. Of course, the fact that all elements are not distorted mentioned by you is important in this case.
Dear Yiming, there is something called reduced integration... but it is not used in this case. In this case the elements were integrated analytical (not even complete Gauss, or any numerical scheme) and hard coded in lines between 91 to 100.
Dear Nicolas, I think line 91 to 99 is just for building stiffness matrix. Since this is pure elastic analysis, no matter how many Guass points used (or analytical solution), the stiffness matrix should be the same, right?
The assembly of the stiffness matrix is done from lines 71 to 78. Lines 91 to 100 are related to local stiffness matrix, in the case of numerical integration this part of the code would change for an integration routine.
The condition of being pure elastic analysis does not determine if the stiffness matrix change or not. The Jacobian would change for distorted elements, and then the function in the integrand. If you change the number of Gauss points, in general, the integral will change (see attached document).
Dear Nicolas,
Yes, you are right, I misunderstood the question. I always thought about the C matrix, which is constant on every Gauss points. But in this case, it's about B matrix. Thank you for your detailed explanation.
Dear Zhang and Zapata,
Thank you for your discussion and detailed answers.
Regards,
Abdullah
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