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Multi-Point Constraints

FrontISTR's multi-point constraint feature provides a means of imposing linear relations \(\sum_i a_i u_i = c\) among multiple nodal degrees of freedom, enabling rigid coupling, periodic boundaries, and arbitrary degree-of-freedom coupling. This chapter summarizes what can be done with multi-point constraints and how to choose between them and other degree-of-freedom coupling features (such as tied contact constraints).

Feature Overview

Multi-point constraints impose the following linear relation on multiple nodal degrees of freedom.

\[ \sum_{i=1}^{N} a_i\, u_i = c \]

A single constraint equation is specified as a combination of the "number of terms \(N\)," "right-hand-side constant \(c\)," and, for each term, "node, degree of freedom, and coefficient \((\text{node}_i, \text{DOF}_i, a_i)\)." Specify at least two terms.

Typical uses are as follows.

  • Rigid coupling: Impose the same displacement on the same degree of freedom of multiple nodes (e.g., connect the \(x\) displacements of two nodes by \(u_x^{(A)} - u_x^{(B)} = 0\)).
  • Periodic boundary: Fix the difference between opposing nodes to 0 or a constant (e.g., \(u^{(A)} - u^{(B)} = c\)).
  • Slave-master coupling: Express a slave degree of freedom as a linear combination of degrees of freedom on the master side.

In the input, define each constraint equation with !EQUATION. Multi-point constraints defined with !EQUATION are effective in structural analyses (static analysis, dynamic analysis, modal analysis, and frequency response analysis) and in heat conduction analyses (steady-state and transient). In structural analysis, translational and rotational degrees of freedom (DOF numbers 1–6) are specified; in heat conduction analysis, the temperature degree of freedom (DOF number 1) is specified.

Choosing Multi-Point Constraints

Choose the method for entering a multi-point constraint according to whether the constrained entities are specified as "individual nodes and degrees of freedom," "surface to surface," or "node to element."

Feature How Constraints Are Specified Main Uses
!EQUATION Directly specify individual nodes, degrees of freedom, and coefficients as an equation Rigid coupling, periodic boundaries, slave-master coupling
!CONTACT INTERACTION=TIED (tied constraint) Couple surfaces specified as a contact pair Permanently couple parts with different mesh densities
!EMBED Couple a group of embedded nodes to a host element group Embed reinforcement in a host material

Use !EQUATION when the constraint relation can be written directly in terms of nodes, degrees of freedom, and coefficients. This is direct for cases such as symmetry and periodic boundaries, where the geometric relation between opposing nodes is known and can be expressed as a linear relation. Use a tied constraint (INTERACTION=TIED) when surfaces with nonmatching meshes are to be coupled, and use embedding when a group of nodes is to be constrained within surrounding elements. Tied constraints and embedding are internally expanded into multi-point constraints and then processed through the same path as !EQUATION.

Sliding contact (INTERACTION=SSLID, FSLID) is a nonlinear constraint whose constraint relation changes according to the contact state, and is not included in the multi-point constraints covered in this chapter. For contact that permits sliding and separation, see Contact and Embedded Elements.

Specifying Constraint Equations

A constraint equation is entered with !EQUATION by specifying the "number of terms \(N\)" and "right-hand-side constant \(c\)," together with, for each term, a combination of "node \(\mathrm{node}_i\)," "degree-of-freedom number \(d_i\)," and "coefficient \(a_i\)." For each term, specify either a node number or a node group name. For the degree-of-freedom number, use translational DOFs 1–3 and rotational DOFs 4–6 in structural analysis, and temperature DOF 1 in heat conduction analysis. A term that specifies a degree of freedom that has no meaning in the model (such as DOFs 4–6 for a solid element with translational DOFs only) is ignored and a warning is output.

If the same node and the same degree of freedom are specified more than once among the terms, the constraint equation becomes degenerate. The second and subsequent duplicate terms are ignored and a warning is output. Within a single constraint equation, specify distinct combinations of node and degree of freedom.

When multiple nodes are specified at once using node groups, the number of nodes in each group must be consistent with the groups used in the other terms. If they are inconsistent, an input error occurs.

The degree of freedom specified in the first term of a constraint equation is treated as the slave degree of freedom during processing. It must not overlap with the first term of another constraint equation, a single-point constraint specified by !BOUNDARY, or a degree of freedom subject to a tied constraint. Constraining the same degree of freedom twice makes the constraint relation redundant and causes the linear system to become ill-conditioned.

For the format, parameters, and order of data lines for a constraint equation, see !EQUATION.

Constraint Handling Methods (Numerical Solution)

The method used to incorporate multi-point constraints into the linear system affects the properties of the coefficient matrix and convergence even when the input constraint equations are the same. FrontISTR provides the penalty method and explicit degree-of-freedom elimination. The default for iterative methods is explicit degree-of-freedom elimination, while the default for direct methods is the penalty method. For selection and specification of the handling method, see the MPC handling methods in Linear Solver and Preconditioning.