长/高比很大的二维有限元模型的求解方法请教

ASCLab_swjtu · 发表于 2011-12-9 17:22:01

问题描述：求解对象是一个二维模型，长度是高度的1000倍左右（高度只有数微米，长度是数毫米）（类似于薄板或者电机中的气隙部分）。就这样一个问题，采用传统的有限元剖分技术，将导致剖分单元异性，进而导致由此形成的系数刚度矩阵病态。针对我面临的问题，请有这方面经验的专家给点有限求解方法和处理病态方程系数矩阵方面的经验和建议，不胜感激啊！

tonnyw · 发表于 2011-12-10 23:04:54

The element with long aspect ratio always causes large error even though we fix the problem of ill-conditioned stiffness matrix. You might try adaptive refinement. Can you upload the finite element model so we all can take a look?

ASCLab_swjtu · 发表于 2011-12-12 16:47:05

Thank you very much your response of my question. I have not yet establish the finite element model up to now. Would you like tell me the real source that causes the difficulty in coping with such a problem with high aspect ratio even though ill-conditioned stiffness matrix has been refined. Thank you in advance!

tonnyw · 发表于 2011-12-14 09:12:04

本帖最后由 tonnyw 于 2011-12-14 09:16 编辑

In Ciarlet's book: The Finite Element Method for Elliptic Problems, he has the mathematical proof about why long aspect ratio can cause large error in the finite element solution.

You really don't need to know the details. You can just take this conclusion as facts.

Here is a link which might be helpful.
http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1453&context=cstech

chentao807 · 发表于 2011-12-14 17:10:10

这个问题就像薄板元不能用实体单元分析一样，如果长宽比比较高的话，就考虑把二维问题转化为一维问题。

ASCLab_swjtu · 发表于 2011-12-21 04:31:38

tonnyw 发表于 2011-12-14 09:12
In Ciarlet's book: The Finite Element Method for Elliptic Problems, he has the mathematical proof ab ...

Thanks a lot for your link which is very useful for proceeding my numerical work! I appreciate very much for your generous help!

ASCLab_swjtu · 发表于 2011-12-21 04:33:06

chentao807 发表于 2011-12-14 17:10
这个问题就像薄板元不能用实体单元分析一样，如果长宽比比较高的话，就考虑把二维问题转化为一维问题。 ...

谢谢您！不过我处理的问题很难转化为一维的。

六月雪 · 发表于 2011-12-21 23:08:20

这个类似结构单元，剪切应力影响转角变化，但对能量积分泛函影响太大，bathe的著作中好像提到过处理方式。

ASCLab_swjtu · 发表于 2012-1-11 20:52:15

谢谢各位的建议。此外，请问，对于含多种媒质的有限元求解域，在进行节点编号时是先编完一个媒质的区域，再转入下一个媒质区域编号，还是根据单元内节点编号相近的原则，对整个求解域进行混合编号呢？

tonnyw · 发表于 2012-1-17 01:51:33

ASCLab_swjtu 发表于 2012-1-11 20:52
谢谢各位的建议。此外，请问，对于含多种媒质的有限元求解域，在进行节点编号时是先编完一个媒质的区域，再 ...

I think it doesn't matter how you enumerate the element or node numbers. After assembling element stiffness matrix, you have to use some algorithm to minimize the band width or just use some sparse solver directly.

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[H. 有限元编程] 长/高比很大的二维有限元模型的求解方法请教