Dual Domain Structural Analysis

The Dual Domain concept was taken up very rapidly by industrial users, so much so that there was immediate pressure on software providers to extend the idea to mold cooling and warpage analysis. The mold cooling analysis was easily extended. The boundary element approach for the mold was retained, and coupled with a heat transfer analysis in the plastic material enclosed by the surface mesh.

In order to permit warpage analysis, a structural analysis capability was first required. Much of the following description was published in [113].

104 6 Numerical Methods for Solution

The method is best introduced by means of an example. The essential idea is to model the structural performance, that is, the bending and membrane characteristics of a plate, using only a surface mesh defining the outer boundaries of the plate. Consider the plate shown in Figure 6.5.

Figure 6.5 A simple plate may be decomposed into two parts, each of half the original thickness, and perfectly bonded together

From the geometric point of view, the flat plate of thickness h can be seen as the perfect bond-ing of two plates each of thickness h/2. If we consider such an assembly, we can see that it could be modeled using two shells, each with their reference surface at the geometric center of the two plates. However, this is problematic, as the nodes defining the mid-surfaces are displaced from the outer surface. We want to use the mesh on the outer surface without mod-ification. This is accomplished by using eccentric shell elements.

As shown in Figure 6.6, a structural shell element may be defined such that its reference plane is located anywhere. The distance from which the reference plane is displaced from the mid-surface is called the eccentricity. Returning to the problem, we can see in Figure 6.5 that the

Figure 6.6 Eccentric shell element for structural analysis

top plate can be modeled using eccentric shell elements with their top surfaces as reference surfaces. Similarly, the bottom plate can be modeled using eccentric shell elements with their bottom surfaces as reference surfaces. This solves the problem by allowing us to use exist-ing nodes on the outer surfaces of the plate. In order to get the correct structural response, however, the two plates of thickness h/2 must be “bonded together” in some way. The bond-ing of the top and bottom plates involves imposbond-ing the Love-Kirchhoff assumption of classical

plate or shell theory [83] and requires that a normal to the plate or shell remains straight after deformation and be unchanged in length. This is accomplished by the use of multi-point con-straints.

In summary, for structural analysis, the Dual Domain method involves the following steps:

■ Meshing of the outer surface of the structure and establishing relationships between ele-ments on the top and bottom surfaces to define local thickness

■ Use of shell elements with their reference surfaces at the surfaces defining the outer bound-ary of the three-dimensional object

■ The use of multi-point constraints to ensure that normals to the top and bottom surfaces remain straight after deformation

The constraints used depend on the type of element chosen. In the past, 3-node triangular el-ements involved constant strain, across the element and their performance was poor. Due to interest in optimization of structures, however, much improved element formulations were de-veloped. One example is a plane triangular facet shell element with 18 degrees of freedom (six at each node—three displacements and three rotations). The element is constructed by super-imposing the local membrane formulation due to Bergan and Felippa [34] with the bending formulation due to Batoz and Lardeur [28] and transforming the combined equations to the global coordinate system. The drilling rotation degree of freedom about a local reference sur-face normal is used in the membrane formulation, and is defined in the local element system by

θz=1 2

µ∂uy

∂x −∂ux

∂y

¶

. (6.1)

To define the relationship between the degrees of freedoms of node n and those of its matching node p (see Figure 6.7), we require that the normals to the mid-surface before deformation remain straight after deformation.

Figure 6.7 Structural elements matched for Dual Domain analysis

Adopting the local coordinate system of the element, we denote the three displacement DOF and the three rotational DOF at node n by uxn, uyn, uznandθxn,θyn,θznrespectively. There are

106 6 Numerical Methods for Solution

then the following relationships between the degrees of freedom of node n and displacements and rotations of its matching point p:

u_xn= uxp− θyph , (6.2)

u_yn= uyp+ θxph , (6.3)

uzn= uzp, (6.4)

θxn= θxp, (6.5)

θy_n= θy_p, (6.6)

and

θzn= θzp+h 2

·∂θxp

∂x +∂θyp

∂y

¸

, (6.7)

where h is the distance between node n and its matching point p. Note that the relationship in Equation 6.7 is obtained using Equation 6.1, and so is particular to the element type chosen.

This system of constraints is imposed at all nodes on the bottom (or top) surface of the model with the exception of those at the edges. Elements forming the edge of the plate are assigned one-sixth of the thickness of the adjacent elements on the top and bottom surfaces.

With these constraints, the structural performance of the composite structure is identical to the original plate. The composite model may now have appropriate boundary conditions and loading applied for structural analysis, and so be used for warpage analysis of the 3D geometry.

In general, the mesh on the top surface is not coincident with the bottom mesh. Hence, a normal from a node n on the bottom surface will not usually coincide with a node on the top surface. Instead, it is more likely that the normal will intersect the top element at a point p (see Figure 6.8). In this case, we interpolate the required constraints using the three nodes defining the top element.

Figure 6.8 Elements on top and bottom surfaces are generally not coincident; that is, the normal from node n of the bottom element, intersects the top element at some point p within the element. In this case, interpolation is required