interface

Specify physical connections between components of mechss model

Syntax

sysCon = interface(sys,c1,nodes1,c2,nodes2)

sysCon = interface(sys,c,nodes)

sysCon = interface(___,KI,CI)

Description

sysCon = interface(sys,c1,nodes1,c2,nodes2) specifies physical couplings between components c1 and c2 in the second-order sparse model sys. nodes1 and nodes2 contain the indices of shared nodes relative to the nodes of c1 and c2. The physical interface is assumed rigid and satisfies the standard consistency and equilibrium conditions. sysCon is the resultant model with the specified physical connections. Use showStateInfo to get the list of all available components of sys.

sysCon = interface(sys,c,nodes) specifies that component c interfaces with the ground. Connecting the node specified in c to the ground amounts to the zero displacement constraint (q = 0).

sysCon = interface(___,KI,CI) further specifies the stiffness KI and damping CI for nonrigid interfaces.

Examples

Physical Connections Between Components in a Sparse Second-Order Model

For this example, consider a structural model that consists of two square plates connected with pillars at each vertex as depicted in the figure below. The lower plate is attached rigidly to the ground while the pillars are attached rigidly to each vertex of the square plate.

Load the finite element model matrices contained in platePillarModel.mat and create the sparse second-order model representing the above system.

load('platePillarModel.mat')
sys = ...
   mechss(M1,[],K1,B1,F1,'Name','Plate1') + ...
   mechss(M2,[],K2,B2,F2,'Name','Plate2') + ...
   mechss(Mp,[],Kp,Bp,Fp,'Name','Pillar3') + ...
   mechss(Mp,[],Kp,Bp,Fp,'Name','Pillar4') + ...
   mechss(Mp,[],Kp,Bp,Fp,'Name','Pillar5') + ...
   mechss(Mp,[],Kp,Bp,Fp,'Name','Pillar6');

Use showStateInfo to examine the components of the mechss model object.

showStateInfo(sys)

The state groups are:

    Type        Name      Size
  ----------------------------
  Component    Plate1     2646
  Component    Plate2     2646
  Component    Pillar3     132
  Component    Pillar4     132
  Component    Pillar5     132
  Component    Pillar6     132

Now, load the interfaced node index data from nodeData.mat and use interface to create the physical connections between the two plates and the four pillars. nodes is a 6x7 cell array where the first two rows contain node index data for the first and second plates while the remaining four rows contain index data for the four pillars.

load('nodeData.mat','nodes')
for i=3:6
   sys = interface(sys,"Plate1",nodes{1,i},"Pillar"+i,nodes{i,1});
   sys = interface(sys,"Plate2",nodes{2,i},"Pillar"+i,nodes{i,2});
end

Specify connection between the bottom plate and the ground.

sysCon = interface(sys,"Plate2",nodes{2,7});

Use showStateInfo to confirm the physical interfaces.

showStateInfo(sysCon)

The state groups are:

    Type            Name         Size
  -----------------------------------
  Component        Plate1        2646
  Component        Plate2        2646
  Component       Pillar3         132
  Component       Pillar4         132
  Component       Pillar5         132
  Component       Pillar6         132
  Interface    Plate1-Pillar3      12
  Interface    Plate2-Pillar3      12
  Interface    Plate1-Pillar4      12
  Interface    Plate2-Pillar4      12
  Interface    Plate1-Pillar5      12
  Interface    Plate2-Pillar5      12
  Interface    Plate1-Pillar6      12
  Interface    Plate2-Pillar6      12
  Interface    Plate2-Ground        6

You can use spy to visualize the sparse matrices in the final model.

spy(sysCon)

The data set for this example was provided by Victor Dolk from ASML.

Input Arguments

sys — Sparse second-order model

mechss model object

Sparse second-order model, specified as a mechss model object. For more information, see mechss.

c — Components to connect

string | array of character vectors

Components of sys to connect, specified as a string or an array of character vectors. Use showStateInfo to get the list of all available components of sys.

nodes — Index information of components to connect

Nc-by-Ni cell array

Index information of components to connect, specified as an Nc-by-Ni cell array, where Nc is the number of components and Ni is the number of physical interfaces.

KI — Stiffness matrix

Nq-by-Nq sparse matrix

Stiffness matrix, specified as an Nq-by-Nq sparse matrix, where Nq is the number of nodes in sys.

CI — Damping matrix

Nq-by-Nq sparse matrix

Damping matrix, specified as an Nq-by-Nq sparse matrix, where Nq is the number of nodes in sys.

Output Arguments

sysCon — Output system with physical interfaces

mechss model object

Output system with physical interfaces, returned as a mechss model object. Use showStateInfo to examine the list of physical interfaces in the system.

Algorithms

Dual Assembly

interface uses the concept of dual assembly to physically connect the nodes of the model components. For n substructures in the physical domain, the sparse matrices in block diagonal form are:

$$\begin{array}{l}M\triangleq diag\left(M_1,\mathrm{...},M_n\right)=\left[\begin{array}{ccc}{M}_1& 0& 0\\ 0& \ddots & 0\\ 0& 0& M_n\end{array}\right]\\ C\triangleq diag\left(C_1,\mathrm{...},C_n\right)\\ \\ K\triangleq diag\left(K_1,\mathrm{...},K_n\right)\\ q\triangleq \left[\begin{array}{c}{q}_1\\ ⋮\\ q_n\end{array}\right],\text{ B }\triangleq \left[\begin{array}{c}{B}_1\\ ⋮\\ B_n\end{array}\right]\text{, F }\triangleq \left[\begin{array}{ccc}{F}_1& \cdots & F_n\end{array}\right],\text{ G }\triangleq \left[\begin{array}{ccc}{G}_1& \cdots & G_n\end{array}\right]\end{array}$$

where, f is the force vector dependent on time and g is the vector of internal forces at the interface.

Two interfaced components share a set of nodes in the global finite element mesh q: the subset N₁ of nodes from the first component coincides with the subset N₂ of nodes from the second component. The coupling between the two components is rigid only if:

• The displacements q at the shared nodes are the same for both components.

  $$q\left(N_1\right)=q\left(N_2\right)$$

• The forces g one component exerts on the other are opposite (by the action/reaction principle).

This relation can be summarized as:

$$M\ddot{q}+C\dot{q}+Kq=Bu+g,Hq=0,g=-H^T\lambda$$

where, H is a localisation matrix with entries 0, 1, or -1. The equation Hq = 0 is equivalent to q(N₁) = q(N₂), and the equation g = −H^T λ is equivalent to g(N₁) = −λ and g(N₂) = λ. These equations can be combined in the differential-algebraic equation (DAE) form:

$$\begin{array}{l}\left[\begin{array}{cc}M& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}\ddot{q}\\ \ddot{\lambda }\end{array}\right]+\left[\begin{array}{cc}C& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}\dot{q}\\ \dot{\lambda }\end{array}\right]+\left[\begin{array}{cc}K& H^T\\ H& 0\end{array}\right]\left[\begin{array}{c}q\\ \lambda \end{array}\right]=\left[\begin{array}{c}B\\ 0\end{array}\right]u\\ y=\left[\begin{array}{cc}F& 0\end{array}\right]\left[\begin{array}{c}q\\ \lambda \end{array}\right]+\left[\begin{array}{cc}G& 0\end{array}\right]\left[\begin{array}{c}\dot{q}\\ \dot{\lambda }\end{array}\right]+Du\end{array}$$

This DAE model is called the dual assembly model of the overall structure. While the principle was explained for two components, this model can accommodate multiple interfaces, including interfaces involving more than two components.

Nonrigid interface

In non-rigid interfaces, the displacements q₁(N₁) and q₂(N₂) are allowed to differ and the internal force is given by:

$$\lambda =K_i\delta +C_i\dot{\delta },\delta \triangleq Hq=q_1\left(N_1\right)-q_2\left(N_2\right)$$

This models spring-damper-like connections between nodes N₁ in the first component and nodes N₂ in the second component. Going from rigid to non-rigid connection eliminates the algebraic constraints Hq = 0 and explicitates the internal forces. Then, eliminate λ to obtain:

$$M\ddot{q}+\left(C+H^T{C}_{i}H\right)\dot{q}+\left(K+H^T{K}_{i}H\right)q=0,y=Fq+G\dot{q}+Du$$

This is the set of the primal-assembly equations for non-rigid coupling form that remains symmetric when the uncoupled model is symmetric. A drawback of this form is that the coupling terms $$H^T{C}_{i}H$$ and $$H^T{K}_{i}H$$ may cause fill-in. To avoid this, interface instead constructs the dual-assembly form:

$$\left[\begin{array}{ccc}M& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}\ddot{q}\\ \ddot{\delta }\\ \ddot{\lambda }\end{array}\right]+\left[\begin{array}{ccc}C& 0& 0\\ 0& C_i& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}\dot{q}\\ \dot{\delta }\\ \dot{\lambda }\end{array}\right]+\left[\begin{array}{ccc}K& 0& H^T\\ 0& K_i& -I\\ H& -I& 0\end{array}\right]\left[\begin{array}{c}q\\ \delta \\ \lambda \end{array}\right]=\left[\begin{array}{c}B\\ 0\\ 0\end{array}\right]u$$