[求助]请教如何测剪应力

二锅头

PADT is pleased to present F. E. Jargon, Professor Ebrius from the Jargon Technical School (JTS). Professor Jargon has kindly volunteered his services as a guest columnist for The Focus. Over the next several months, Prof. Jargon will be introducing common finite element terms and explaining their meaning and relevance to the finite element engineer.

Greetings, students! I am honored to have the opportunity to write for such a prestigious periodical. It pleases me to take a break from my jargon teachings to share my knowledge with you.

One term that may puzzle you is the “ill-conditioned matrix.” You sometimes get a warning when using the iterative solvers stating that the stiffness matrix may be ill-conditioned with a suggestion to use the sparse solver. This may cause you to think, “Wow, they seem to use that term so nonchalantly, as though I should have learned it in first grade. Did I have chicken pox that week?”

You may not have heard it in first grade, but the term may have reared its head in your linear algebra class. Perhaps you were too eager to get to the “real” engineering courses to have remembered. A matrix is ill-conditioned if small changes in the coefficients of the solution have drastic effects on the results, which makes iterating the solution to a small residual a tricky operation. Numerical round-off in the system can also present some challenges when it comes to solving a model having an ill-conditioned matrix.

An ill-conditioned matrix can represent the equations of two lines that are almost parallel, but not quite. Consider the following system of equations, written in the form of [K]{x}={F}:



The answers are x1 = x2 = 1. But, round off 6.0001 to 6 and we get x1 = 3 and x2 = 0. Leave the 6.0001 number alone and round off 4.0001 to 4 and we get two conflicting equations. Round off both 4.0001 and 6.0001 to 4 and 6 and we end up with two redundant equations. Just tampering with the values slightly has resulted in radically different solutions!

Another type of ill-conditioned matrix — and one more relevant to the finite element analyst — is when we have [K] matrix values that vary by several degrees of magnitude. Like the “near parallel” example, this can also result in widely discrepant results due to small perturbations in the load values. This scenario is caused by radical differences in stiffnesses (or conductivities, etc.) within the model. This sometimes happens in shell models where the tensile stiffnesses are much higher than the rotational stiffnesses. It can also occur in a model having two materials whose Young’s moduli differ by several orders of magnitude. I’ve also seen this in 3D models that have a large amount of bending flexibility compared to the overall tensile stiffnesses. Short beam elements having a much higher stiffness than other elements in the model can also cause ill-conditioning. Poor meshes having elements with large aspect ratios or Jacobians can contribute to this phenomenon as well.

The ill-conditioned matrix is particularly troublesome when trying to use the iterative solvers. Because ANSYS is guessing at an initial solution and iterating to a certain tolerance value, the divergent nature of the ill-conditioned solution makes the iterative process inefficient compared to using a direct solver. However, a slow solution is only a minor inconvenience, comparatively speaking. Because of the potential for discrepant results, it is quite possible to come up with results that are just plain wrong, regardless of the solver used. For now, the only way to avoid this is to be aware of the pitfalls and avoid them. The skilled ANSYS user will take extra steps to achieve a robust mesh and verify the accuracy of his or her results.

Ever run across a word in the ANSYS world that you were unsure about but were afraid to ask about? Send us an e-mail and we will forward it to Professor Jargon. All submissions will be kept strictly anonymous.

References

Gene Poole, Technical Fellow, ANSYS Inc.

Chapra, Steve C. and Raymond P. Canale. Numerical Methods for Engineers. 2nd Ed. New York: McGraw Hill, 1988.

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