The Finite Difference Method - Mechanics of Structures: Variational and Computational Methods - page 516

Fig. 8.9c can be expressed as

φ
2 h

φ
2 h

φ ∂ψ
∂

G

φ ∂ψ
∂

G

τ = G

z =

(ψ − ψ ) τ =− G

y =−

(ψ − ψ ) (8)

xy 3 , 4 3 , 2 xz 4 , 3 2 , 3

8.2.2 Variable Mesh Size
If the region of concern is irregular in shape so that it would be difficult to base the finite
difference approximations on a uniform mesh, special accommodation can be made. Stan-
dard references on finite differences describe methods of creating nonuniform meshes. For
example, a Taylor series can be used to assist in accounting for a point on a boundary curve
that intersects the mesh at other than a grid point. The integral-based finite differences
procedure of Section 8.3 is very convenient for approximating differential equations for
irregularly shaped regions with a nonuniform mesh.

8.3 Variationally Based Finite Differences

As we have observed, the conventional finite difference equations are formed by replacing
the derivatives of the variables in the governing differential equations by their difference
quotients. This leads to a system of linear algebraic equations for the values of the variables
at the mesh points. Such finite difference formulations are usually restricted to a regular
mesh because, in most instances, an irregular mesh leads to a nonsymmetric matrix for
the linear system of equations. Furthermore, considerable difficulty can occur with the
conventional finite difference method in incorporating the boundary conditions. Not only
are special procedures, such as the use of fictitious points, sometimes necessary, but all
boundary conditions, i.e., both the force and displacement conditions, regardless of the
topology on which they occur must be approximated by the differencing scheme. In recent
years, it has been demonstrated that finite difference approximations can also be derived
from a variational approach if the derivatives of the variables in the variational functional
are replaced by corresponding difference quotients [Brush and Almroth, 1975; Bushnell,
1973; Bushnell and Almroth, 1971; Griffin and Kellogg, 1967; Griffin and Varga, 1963]. The
variational principle then leads to a system of linear algebraic equations for the variables at
the mesh points. Symmetric matrices can be generated because the equations are derived
from a variational functional. Another advantage of the variational approach is that it may
not be necessary to enforce some of the boundary conditions, since often not all of the
boundary conditions are included in the variational formulation.
To be more specific about a variationally based finite differences approach, which is
sometimes called the finite difference energy method , consider the use of the principle of
virtual work. The derivatives in the integrals for internal and external work are replaced
by difference expressions. Since the integrands are then piecewise constant, the integrals
are easily evaluated, especially by the integration procedures sometimes employed with
finite element calculations. The principle of virtual work, 0, leads to a system of

linear algebraic equations for the unknown displacements at the nodes. Since invoking the
principle of virtual work implies that the conditions of equilibrium and the force boundary
conditions will be satisfied as well as possible, the algebraic difference equations must
satisfy only the displacement boundary conditions. This is in contrast to the usual finite

δ W =