Find the stiffnessmatrix for piecewire quadrtic basis functi

Find the stiffnessmatrix for piecewire quadrtic basis function integral_^1 dx

Solution

Let us calculate the specific matrix entry A2,3=23dx. Figure Illustration of the piecewise linear basis functions corresponding to global node 2 and 3 shows how 2 and 3 look like. We realize from this figure that the product 230 only over element 2, which contains node 2 and 3. The particular formulas for 2(x) and 3(x) on [x2,x3] are found from (3). The function 3 has positive slope over [x2,x3] and corresponds to the interval [xi1,xi] in (3). With i=3

we get

3(x)=(xx2)/h,

while 2(x)

has negative slope over [x2,x3] and corresponds to setting i=2

in (3),

2(x)=1(xx2)/h.

We can now easily integrate,

A2,3=23dx=x3x2(1xx2h)xx2hdx=h6.

The diagonal entry in the coefficient matrix becomes

A2,2=x2x1(xx1h)2dx+x3x2(1xx2h)2dx=h3.

The entry A2,1

has an the integral that is geometrically similar to the situation in Figure Illustration of the piecewise linear basis functions corresponding to global node 2 and 3, so we get A2,1=h/6

.

Calculating a general row in the matrix

We can now generalize the calculation of matrix entries to a general row number i

. The entry Ai,i1=ii1dx involves hat functions as depicted in Figure Illustration of two neighboring linear (hat) functions with general node numbers. Since the integral is geometrically identical to the situation with specific nodes 2 and 3, we realize that Ai,i1=Ai,i+1=h/6 and Ai,i=2h/3

. However, we can compute the integral directly too:

Ai,i1=ii1dx=xi1xi2ii1dxi=0+xixi1ii1dx+xi+1xiii1dxi1=0=xixi1(xxih)i(x)(1xxi1h)i1(x)dx=h6.

The particular formulas for i1(x)

and i(x) on [xi1,xi] are found from (3): i is the linear function with positive slope, corresponding to the interval [xi1,xi] in (3), while i1 has a negative slope so the definition in interval [xi,xi+1] in (3) must be used. (The appearance of i in (3) and the integral might be confusing, as we speak about two different i indices.)