How Can the Master–Slave Elimination for Multi‐Point Constraints be Drastically Accelerated?

Source abstract

ABSTRACT Multi‐point constraints are essential in modeling various engineering problems, for example, in the context of joints undergoing large rotations or coupling of different element types in finite element analysis. The master–slave elimination is an efficient method for the numerical treatment of such constraints because it reduces the dimension of the resulting linear system. However, the method requires the computation of the reduced row echelon form of a submatrix of the constraint Jacobian. For nonlinear constraints, this computation has to be performed at every iteration step of the Newton–Raphson scheme. Although this process is computationally expensive for large systems, the method exhibits a reduced computational complexity compared to multiplier and the penalty methods, nevertheless. Therefore, a method for drastically increasing the computational efficiency of this already efficient method is presented. It is based on the exploitation of the specific structure of the constraint Jacobian as it appears in typical engineering applications. The analysis of this structure is based on the identification of constraint clusters, which can be performed in a preprocessing step without significant computational effort. All matrix operations required for this calculation are performed using the compressed sparse row (CSR) technique for storing sparse matrices, which is also particularly advantageous because all the matrices required for this are already stored in this format. Finally, the master–slave elimination is extended by the exploitation of the structure of the constraint Jacobian, which reduces the computational costs. The speed‐up of the improved master–slave elimination over the previous master–slave elimination is demonstrated using a numerical example.