Nonlinear finite elements/Axial bar approximate solution

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[edit] Approximate Solution: The Galerkin Approach

To find the finite element solution, we can either start with the strong form and derive the weak form, or we can start with a weak form derived from a variational principle.

Let us assume that the approximate solution is \mathbf{u}_h(\mathbf{x}) and plug it into the ODE. We get

AE \cfrac{d^2\mathbf{u}_h}{dx^2} + a \mathbf{x} = R_h(\mathbf{x})

where Rh is the residual. We now try to minimize the residual in a weighted average sense

\int_0^L R_h(\mathbf{x}) \mathbf{w}(\mathbf{x}) ~dx = 0

where \mathbf{w}(\mathbf{x}) is a weighting function. Notice that this equation is similar to equation (5) (see 'Weak form: integral equation') with \mathbf{w} in place of the variation \delta \mathbf{u}. For the two equations to be equivalent, the weighting function must also be such that \mathbf{w}(0) = 0.

Therefore the approximate weak form can be written as

{ \int_0^L AE \cfrac{d\mathbf{u}_h}{dx}\cfrac{d\mathbf{w}}{dx}~dx = \int_0^L\mathbf{q}\mathbf{w}~dx + \left. \boldsymbol{R}~\mathbf{w}\right|_{x=L} ~. }

In Galerkin's method we assume that the approximate solution can be expressed as

\mathbf{u}_h(\mathbf{x}) = a_1\varphi_1(\mathbf{x}) + a_2\varphi_2(\mathbf{x}) + \dots + a_n\varphi_n(\mathbf{x}) = \sum_{i=1}^n a_i \varphi_i(\mathbf{x}) ~.

In the Bubnov-Galerkin method, the weighting function is chosen to be of the same form as the approximate solution (but with arbitrary coefficients),

\mathbf{w}(\mathbf{x}) = b_1\varphi_1(\mathbf{x}) + b_2\varphi_2(\mathbf{x}) + \dots + b_n\varphi_n(\mathbf{x}) = \sum_{j=1}^n b_j \varphi_j(\mathbf{x}) ~.

If we plug the approximate solution and the weighting functions into the approximate weak form, we get

\int_0^L AE \left(\sum_{i=1}^n a_i\cfrac{d\varphi_i}{dx}\right) \left(\sum_{j=1}^n b_j\cfrac{d\varphi_j}{dx}\right)~dx = \int_0^L\mathbf{q}\left(\sum_{j=1}^n b_j\varphi_j\right)~dx + \left. \boldsymbol{R}~\left(\sum_{j=1}^n b_j\varphi_j\right)\right|_{x=L} ~.

This equation can be rewritten as

\sum_{j=1}^n b_j \left[\int_0^L AE \left(\sum_{i=1}^n a_i\cfrac{d\varphi_i}{dx} \cfrac{d\varphi_j}{dx}\right)~dx \right] = \sum_{j=1}^n b_j \left[\int_0^L\mathbf{q}\varphi_j~dx + \left. \left(\boldsymbol{R}~\varphi_j\right)\right|_{x=L}\right] ~.

From the above, since bj is arbitrary, we have

\int_0^L AE \left(\sum_{i=1}^n a_i\cfrac{d\varphi_i}{dx} \cfrac{d\varphi_j}{dx}\right)~dx = \int_0^L\mathbf{q}\varphi_j~dx + \left. \boldsymbol{R}~\varphi_j\right|_{x=L} ~, ~ j = 1\dots n.

After reorganizing, we get

\sum_{i=1}^n \left[\int_0^L \cfrac{d\varphi_j}{dx} AE \cfrac{d\varphi_i}{dx}~dx\right] a_i = \int_0^L \varphi_j\mathbf{q}~dx + \left. \varphi_j\boldsymbol{R}\right|_{x=L} ~, ~ j = 1\dots n

which is a system of n equations that can be solved for the unknown coefficients ai. Once we know the ais, we can use them to compute approximate solution. The above equation can be written in matrix form as

\mathbf{K} \mathbf{a} = \mathbf{f} \qquad \leftrightarrow \qquad K_{ji} a_i = f_j

where

\mathbf{K} = \int_0^L \mathbf{B}^T \mathbf{D} \mathbf{B}~ dx \qquad \leftrightarrow \qquad K_{ji} = \int_0^L \cfrac{d\varphi_j}{dx} AE \cfrac{d\varphi_i}{dx}~dx

and

f_j = \int_0^L \varphi_j\mathbf{q}~dx + \left. \varphi_j\boldsymbol{R}\right|_{x=L}~.

The problem with the general form of the Galerkin method is that the functions \varphi_i are difficult to determine for complex domains.

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