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The Stable Finite Element Method for Minimization Problems

Bin Liu's picture

The conventional finite element method is difficult to converge for a non-positive definite stiffness matrix, which usually occurs when the material displays softening behavior or when the system is near the state of bifurcation.  We have developed two stable algorithms for a non-positive definite stiffness matrix, one for the direct linear equation solver and the other for the iterative solver in the finite element method for minimization problems.  For a direct solver with non-positive definite stiffness matrix, energy minimization of a system with multiple degrees of freedom (DOF) is decomposed to the minimization of many 1-DOF systems, and for the latter an efficient and robust minimization method is developed to ensure that the system energy decreases in every incremental step, regardless of the positive definiteness of the stiffness matrix.  For an iterative solver, the stiffness matrix is modified to ensure the convergence, and the modified stiffness matrix indeed leads to the correct solution.  An example of a single wall carbon nanotube under compression is studied via the proposed algorithms.

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Comments

Jianliang Xiao's picture

Bin,

It's a very clever idea to decompose a complex system into many 1 DOF systems, which makes the problem much easier. Good to know that!

I have a small question, how about the cases if the search of energy minimum encountered a point with P=0 (dE/dx=0) and K<0 (local energy maximum) or K=0 (saddle point)?

Bin Liu's picture

Jianliang,

Thanks for your positive comments. Actually, the case you mentioned, P=0 (dE/dx=0) and K<0 (local energy maximum) or K=0 (saddle point), has almost no possibility to emerge in numerical computation because it is very difficult to obtain accurate zero after computation on a computer due to the numerical error, although it is possible from theoretical point of view. In the case of K=0 (saddle point), one may treat it in the same way as K<0. For P=0 (dE/dx=0) and K<0 (local energy maximum), one may add an initial small perturbation to trigger the minimization process, and in most times the computer can automatically introduce this perturbation by a numerical error.

Jianliang Xiao's picture

Bin,

Got it! Thanks for your answer!

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