Tags » Solution

有限元求解方法若干

基本方程

不失一般而言,对于一个含有个自由度的问题中,其中是未约束自由度的个数,是约束自由度的个数,在形成了整体刚度矩阵之后,需要求解以下方程

,

其中,是阶数为的整体刚度矩阵,是位移向量,是对应的荷载向量。

由于个约束自由度的位移值已经由位移边界条件给定,在求得位移向量后,这个约束自由度上的反力也可以将代入原方程后求得,故通常我们不将这个约束自由度上的位移或反力作为未知量。

这样,我们只考虑一个阶的问题,位移向量和荷载向量相应写作和。

再考虑荷载条件,假设给定了个外力荷载条件,则在中,有个分量为给定值,其余个分量由于未被约束,应均为零值。这样,整个方程中在给定了个力荷载分量后有个位移值需要求解

如果给定了个位移荷载条件,同样的,在中,有个分量为给定值,其余个分量均为未知待求量。在相应的荷载向量中,所对应的个分量为未知量,其余个分量均为零值。这样,整个方程中在给定了个位移荷载分量后有个位移值和个反力值需要求解

如果我们将给定的荷载条件视为额外引入方程的约束条件,而将对应的位移值或力值视为未知量,则实质上,对于原来的阶方程,在原来的以个位移边界条件作为约束(以避免刚体位移)的基础上,我们引入了个额外的约束条件,来求解一个阶的问题。

当然,如果我们仅仅以个位移边界条件求解原先的阶方程,将得到零状态,也即初始状态。

简单起见,以下我们讨论的情况,这样,原先的问题就变成了一个阶的问题,在这基础上的讨论另可参见Yang and Shieh, 2009

Force-Controlled

基于力荷载控制的增量迭代算法是最常用的求解算法之一,通常给出的约束条件是

,

也即给定第个自由度的外力值,考虑到数值稳定性,我们往往采用荷载因子乘以参考荷载的形式定义外荷载大小,即

, 10 more words

Solution

Metaphorical Shelter From The Hypothetical Storm (6 min read) — The Millionaire's Digest

Written by Millionaire’s Digest Team Member: Noelle L. Reagan Founder& Owner of: Made Up by Reagan Millionaire’s Digest Team, Contributor, Successful Living and Writing Writer Dating after the damage of sheer and almost unbearable heartbreak.

37 more words
Re-blogs

E-SPIN Unified Vulnerability Management (UVM) Explained

E-SPIN Unified Vulnerability Management (UVM), the company use to unify entire solution portfolio across multiple specialization domain is being explained during one of the E-SPIN event session. 58 more words

Solution

Do you take short cuts when telling your story?

While our clients vary from sole traders to multinationals, selling anything from chocolate bars to big tech installations, they’re all confronted by a similar task. They need to tell the story of their product or service. 508 more words

NC

Azure Active Directory Labs Series – Multi-Factor Authentication

Back in June I had the pleasure of delivering a training on Azure Active Directory to two customer crowds. I say pleasure because not only do I love to share knowledge but also, the preparation of the training forces me to go deep on some aspects of what I’m going to teach. 540 more words

Solution