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:

$$\begin{array}{l}M\triangleq diag\left(M_1,\mathrm{...},M_n\right)=\left[\begin{array}{ccc}{M}_1& .& .\\ .& \ddots & .\\ .& .& M_n\end{array}\right]\\ C\triangleq diag\left(C_1,\mathrm{...},C_n\right)\\ \\ K\triangleq diag\left(K_1,\mathrm{...},K_n\right)\\ q\triangleq \left[\begin{array}{c}{q}_1\\ ⋮\\ q_n\end{array}\right],f\triangleq \left[\begin{array}{c}{f}_1\\ ⋮\\ f_n\end{array}\right],g\triangleq \left[\begin{array}{c}{g}_1\\ ⋮\\ g_n\end{array}\right]\end{array}$$

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

In the concept of dual assembly, the global set of degrees of freedom (DoFs) q is retained and the physical coupling is expressed as consistency and equilibrium constraints at the interface. For rigid connections, these constraints are of the form:

$$Bq=0,g=-B^T\lambda$$

where g is the vector of internal forces at the interface, and the matrix B is permutable to [I -I]. For a pair of matching nodes with indices i₁,i₂ where i₁ selects a node in the first component while i₂ selects the matching node in the second component, $$Bq=0$$ enforces consistency of displacements

$$q\left(i_1\right)=q\left(i_2\right)$$

while $$g=-B^T\lambda$$ enforces equilibrium of the internal forces g at the interface:

$$g\left(i_1\right)+g\left(i_2\right)=0$$

Combining these constrains with the uncoupled equations $$M\ddot{q}+C\dot{q}+Kq=f+g$$ leads to the following dual assembly model for the coupled system:

$$\left[\begin{array}{cc}M& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}\ddot{q}\\ \lambda \end{array}\right]+\left[\begin{array}{cc}C& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}\dot{q}\\ \lambda \end{array}\right]+\left[\begin{array}{cc}K& B^T\\ B& 0\end{array}\right]\left[\begin{array}{c}q\\ \lambda \end{array}\right]=\left[\begin{array}{c}f\\ 0\end{array}\right]$$

Nonrigid interface

Non-rigid interfaces are expressed in the following form:

$$Bq+\delta =0,g=B^T\left(K_c\delta +C_c\dot{\delta }\right)$$

This models a spring-damper connection between two matching nodes at the interface and corresponds to the internal force $$\lambda =K_c\delta +C_c\dot{\delta }$$. In DAE form, it can be rewritten as:

$$\left(\begin{array}{cc}I& 0\\ 0& 0\end{array}\right)\frac{d}{dt}\left(\frac{\delta }{\dot{\delta }}\right)=\left(\begin{array}{cc}0& I\\ I& 0\end{array}\right)\left(\frac{\delta }{\dot{\delta }}\right)+\left(\begin{array}{c}0\\ B\end{array}\right)q,g=B^T\left(K_c{C}_c\right)\left(\frac{\delta }{\dot{\delta }}\right).$$

Note that eliminating $$\delta ,\dot{\delta }$$ amounts to replacing the aggregate C,K matrices by $$C+B^T{C}_{c}B,K+B^T{K}_{c}B$$ using

$$M\ddot{q}+C\dot{q}+Kq=f+g,g=-B^T{K}_{c}Bq-B^T{C}_{c}B\dot{q}$$

which shows how non-rigid connections modify the overall damping and stiffness.