Lecture 14

Finite Element Models

Anderson and Woessner, Sections 3.3, Box 3.3

Wang and Anderson, Chapter 6

Finite element models are more flexible in designing a grid.

We have discussed the types of elements and how they can easily deform to match
irregularly shaped boundaries.

Finite element grids have no inactive nodes.

It is critical to approximate the boundary as close as possible.

A disadvantage of finite elements grids is the larger amount of data required for the grid.

Must have numbered nodes and elements, consecutively.

Must specify the coordinates of each node and element.

Must specify numbering of nodes systematically from top to bottom and from
left to right, sequentially across the shortest dimension of the problem domain.

See figure 14-1a and 14.1b.

Model assembles a global matrix equation from equations for each element.

The sequential numbering minimizes the bandwidth and thus computer storage.

Semibandwith

Must be a symmetric matrix (entries reflect across the diagonal)

SBW = maximum number of columns between the diagonal and the last
nonzero entry, inclusive, along any row of the matrix.

SBW = R + 1

where

R = maximum difference in any two node numbers
that define a single element within the grid.

Bandwidth

Bandwith = 2 x (SBW - 1)

In figure 14-1,

R = 17, SBW = 18 and therefore, Bandwidth = 35.

 

Aspect ratio

	In homogeneous materials, construct elements with aspect ratio near unity.
	Aspect ratio is the maximum to minimum element dimensions.
	Similar to the 1.5 factor in adjacent finite-difference cell dimensions.
	Therefore, can minimize numerical error by using equilateral triangular elements.
	Avoid aspect ratios greater than 5.

Always change adjacent element sizes gradually.

For all of these reasons, it is very difficult to create and modify a finite-element model
without a pre-processor.

Galerkin's Method

Probably the most common solution of finite-element models.

Based on a particular weighted residual principle which is equivalent to a variational principle.

Variational principle

Minimize a physical quantity over a problem domain.

Use potentials (hydraulic head) as the physical quantity to minimize.

Dissipation of energy loss during ground-water flow is at a minimum.

Use the potential energy in ground-water to derive mathematics

The potential is a force potential.

where

F_x and F_y = components of force per unit mass of water.

Remember, force per unit volume is n_wF_x.

Also remember that the average linear ground-water flow velocity is

Then at each point in the ground-water flow system, the rate of energy loss per unit of
aquifer volume is the scalar product of the force per unit aquifer volume times the average
linear ground-water flow velocity.

If we then integrate this over the aquifer volume,

Substitute Darcy's law, then the Dissipation J is

where

b = aquifer thickness.

This simplifies if we assume density, gravity and thickness are constant.

Leads to algebraic equations.

We now can apply Galerkin's method to Laplace's equation.

1) Define an approximate or trial solution to hydraulic head at x, y.

Express as a series summation, or

each term is a product of a nodal head h_L and an associated nodal basis function N_L(x,y).

where

L = nodal number, and

NNODE = total number of nodes in problem domain.

The basis functions are analogous to unit vectors.

The trial solution is built up as a linear combination of the basis functions.

The basis functions can also be called interpolation functions.

They define the trial solution.

2) Define NNODE conditions to calculate NNODE values of h_L.

With the Galerkin method, assume that the starting condition is that the
beginning residuals are zero when integrated over the entire problem
domain.

where

D = integration over entire problem domain.

The terms in the ( ) represent the residual.

The residual is a measure of how poorly (or well) the trial solution satisfies Laplace.

N_L(x, y) then becomes a weighting function.

3) Integrate the basis functions in a piecewise manner (by parts).

4) Define the trial solution by linearly interpolating the nodal values at all three nodes on the
triangular element.

Develop element interpolation (basis) functions for each node of the element.

Therefore, we get a series of algebraic equations represented as basis functions of
interpolated values of trial hydraulic heads.

5) Develop a conductance matrix that consists of conductance values for each element.

Matrix is symmetric.

6) Incorporate boundary conditions.

7) Iteratively solve using such techniques as Gauss-Seidel.

FEMWATER uses the Galerkin finite element method.

	Originally written by G. Yeh.
	Works in variably saturated media and saturated media.

AQUIFEM directly solves using Crout's method

	A fully saturated 2-D flow model.
	3-D code is now available.

Finite element models have the same types of boundary conditions applied as finite-difference models.

	Specified head, specified flow, and head-dependent.
	Can handle more complex shapes and sizes of boundaries more accurately.

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