## 1. Introduction

Shell-to-solid coupling (SSC) is necessary when the discontinuity of the interpolated displacements between adjacent shell and solid elements must be minimized[9]. This scenario is typically found in global-local analyses where the global part of the model consists of shell elements while the local part consists of solid elements. Usually, the local part is meshed much finer.

This section describes the SSC elements and how they can be generated automatically.

Alternatively, the field_transfer method can be employed for shell-to-solid coupling. While the SSC elements constitute a point-wise coupling, the field_transfer command represents a weighted-residual coupling.

## 2. Requirements and implementation

In addition to different element sizes and node positions, the shell-to-solid coupling must also account for the transformation of the rotational degrees-of-freedom of the shell elements to the translation degrees-of-freedom of the connected solid elements. This coupling depends on how the shell element directors are calculated from the degrees-of-freedom, a procedure which is specific to the shell element's formulation and implementation. Hence, the shell-to-solid coupling must be tailor-made for each type of shell element.

In geometrically nonlinear analysis, large rigid-body motions may occur. The shell-to-solid coupling must be performed such that the coupling between the degrees-of-freedom is performed taking into account the current (rotated) configuration. Hence, a linear approach such as enforcing minimal displacement discontinuity by means of the linc command will fall short of this task.

For these reasons, shell-to-solid coupling in B2000++ is implemented in the form of Finite Elements which (a) are specifically made for each supported shell element type, and (b) support geometrically nonlinear analysis. This allows for incorporating the coupling in the standard solution procedures involving the computation of the first and second variation.

The shell-to-solid coupling elements for the MITC shell elements in B2000++ enforce their constraints using B2000++'s constraint system. By default, the Augmented Lagrange method is used for the static linear and nonlinear solvers, and the Lagrange method is used for the dynamic nonlinear solver. See Imposing Constraints for details.

## 3. Automatic definition of SSC elements

When the add_ssc_elements command is specified in the adir command, the B2000++ input processor b2ip++ will add the necessary SSC elements automatically. This way, the explicit definition of SSC elements in the MDL file is not necessary. Example:

adir
case 1
end

The criteria defining when and how these elements are added are based on the evaluation of geometrical properties; in particular, the volume element's node must lie (within sufficient tolerance) in the plane defined by the shell element's edge. Thus, this automatic procedure is limited to straight and moderately curved shell elements.

Instead of having the SSC elements added automatically, the SSC elements also can be specified explicitly in the MDL file. This is described in the following.

## 4. SSC element connectivity

The shell-to-solid coupling elements are rigid and have no materials associated to them. They connect a single edge of a shell element with a single node of 3D solid element. Hence, all nodes of all adjacent solid element's faces must have a separate coupling element associated.

In the above figure, a shell element defined by nodes n1, n2, n3, and n4 is to be linked to a solid element face defined by nodes a, b, c, and d. The link is defined such that the solid element face (green) deforms in the same plane as the shell edge (defined by nodes n1 and n2 in the above figure), the shell edge deforming according to the shell node directors (dashed blue lines). Thus, for the above configuration, 4 coupling elements must be defined, each of them connecting the shell nodes n1-n4 to one of the solid face nodes 1-4:

 n1 n2 n3 n4 a n1 n2 n3 n4 b n1 n2 n3 n4 c n1 n2 n3 n4 d

The element node connectivity sequence of all coupling elements are specified with first specifying the nodes defining the shell followed by one of the solid element nodes:

 n1 n2 ... nnne nk

n1 to nnne are the nodes defining the shell element according to the element connectivity definition of the specific shell element, with the following modification: n1 and n2 are nodes defining the shell element edge to be coupled to the solid. This means that in some cases the original shell element connectivity and the connectivity can be permuted! nk is the k-th element node connectivity index of the solid element involved in the coupling process.

## 5. Element types

Shell-to-solid coupling element types are available for all MITC shell elements.

 SSC.T3.S.MITC Shell-to-solid coupling element linking a T3.S.MITC triangular shell element to a single node of a 3D solid element. The solid element node will deform with the edge of the shell element defined by the first 2 nodes in the element connectivity list. SSC.T6.S.MITC Shell-to-solid coupling element linking a T6.S.MITC second-order triangular shell element to a single node of a 3D solid element. The solid element node will deform with the edge of the shell element defined by the first 2 nodes in the element connectivity list. SSC.Q4.S.MITC Shell-to-solid coupling element linking a Q4.S.MITC quadrilateral shell element to a single node of a 3D solid element. The solid element node will deform with the edge of the shell element defined by the first 2 nodes in the element connectivity list. SSC.Q8.S.MITC Shell-to-solid coupling element linking a Q8.S.MITC second-order quadrilateral shell element to a single node of a 3D solid element. The solid element node will deform with the edge of the shell element defined by the first 2 nodes in the element connectivity list. SSC.Q9.S.MITC Shell-to-solid coupling element linking a Q9.S.MITC second-order quadrilateral shell element to a single node of a 3D solid element. The solid element node will deform with the edge of the shell element defined by the first 2 nodes in the element connectivity list.

## 6. Example of SSC definition in the MDL file

The following, simple example demonstrates how SSC elements can be used. It consists of a single quadrilateral shell element which is to be connected to a single hexahedral solid element. The thickness of the shell element is equal to that of the solid element, and the positions in the x- and y-direction at the interface coincide. Note that this is not required, there may be multiple solid elements in the thickness and also multiple solid elements in the y-direction.

The SSC elements are displayed as blue lines going from the node of the 3D solid element to the centre of the shell element.

title "2-element SSC example"

nodes
1  -1.0  0.0  0.0
2   0.0  0.0  0.0
3   0.0  1.0  0.0
4  -1.0  1.0  0.0
11   0.0  0.0 -0.1
12   1.0  0.0 -0.1
13   1.0  1.0 -0.1
14   0.0  1.0 -0.1
15   0.0  0.0  0.1
16   1.0  0.0  0.1
17   1.0  1.0  0.1
18   0.0  1.0  0.1
end

elements
type Q4.S.MITC mid 1 thickness 0.2
1  1 2 3 4
type HE8.S.TL mid 1
2  11 12 13 14 15 16 17 18
type SSC.Q4.S.MITC
3  2 3 4 1 11
4  2 3 4 1 14
5  2 3 4 1 15
6  2 3 4 1 18
end

material 1 type isotropic
e 73.1e3
nu 0.3
end

ebc 1
dof [UX UY UZ RX RY RZ]  value 0.0  nodes [1 4]
end

nbc 1
dof FZ  value 1.0  nodes [12 13 16 17]
end

case 1
analysis  linear
ebc       1
nbc       1
end

end