Stiffness Matrix

[Prashanth]

Hey Everone,

I have recently started studying FEM and I have some basic doubts. Hope I can get some clarity here.

 

1. Is the stiffness matrix defined for element or node? In a simple truss with 2 linear elements, we calculate the stiffness matrix for K(1,1), K (1,2), K(2,1), K(2,2). In K(e,n): e is the element number and n is the node number. So is the stiffness matrix defined for node or element?

2. How is the order of Stiffness matrix calculated? 

3. Does the stiffness matrix only depends on geometry and material? are there any other factors?

4. Why is the load vector not symmetrical? In a simple truss with 2 linear elements, why is F(1,2) not equal to F(2,1)?  F(e,n): e is the element number and n is the node number.

Thanks and Regards,

Prashanth Setty

[Wolfgang Bangerth]

On 8/9/22 06:53, Prashanth wrote:
> 1. Is the stiffness matrix defined for element or node? In a simple truss with
> 2 linear elements, we calculate the stiffness matrix for K(1,1), K (1,2),
> K(2,1), K(2,2). In K(e,n): e is the element number and n is the node number.
> So is the stiffness matrix defined for node or element?

Prashant,
you seem to be stuck on a very specific notation that I assume you have from a
book or course you took. Since I don't know what the source of that notation
is, it's hard for me to tell you what K(2,1) etc corresponds to -- you'll have
to check how your reference defines this.

But you may want to take a look at step-3 and step-4, for example, where we
first build a dense representation of the stiffness matrix contribution of a
specific element, say A_K(i,j) where K is the element and i,j are local
indices of the degrees of freedom that live on that cell. And then we add
these local contributions into the global stiffness matrix A(i,j) where i,j
are global indices of degrees of freedom.

(We number indices starting at zero, as is common in C++. You may need to take
that into account when translating your notation.)

> 2. How is the order of Stiffness matrix calculated?

There is an enumeration of all degrees of freedom. How they are enumerated is
not particularly important.

> 3. Does the stiffness matrix only depends on geometry and material? are there
> any other factors?

It depends on the differential operator.

> 4. Why is the load vector not symmetrical? In a simple truss with 2 linear
> elements, why is F(1,2) not equal to F(2,1)?  F(e,n): e is the element number
> and n is the node number.

Why should it?

Best
W.

--
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Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/