VCP

Overview

The Variable Condensation Preconditioner (VCP) is designed to condense out variables from the linear system of equations and apply the preconditioner/solver on the reduced simplified system of equations. The development of VCP is motivated by the need to enable a broader range of robust and scalable preconditioners for problems that have a saddle point type of Jacobian. The saddle point type of Jacobian may come from different applications. One typical example is the enforcement of constraints using Lagrange multipliers. Its special numerical character prevents the usage of many scalable iterative solvers. To resolve this issue, a static condensation step is carried out in VCP to remove the Degree of Freedoms (DoFs) that are associated with the Lagrange multipliers. This may result in an easy-to-solve system (sometimes it is positive definite ) which can be handled by a broader range of solvers/preconditioners with improved efficiency. With VCP, we can efficiently apply iterative solvers, e.g., BoomerAMG, to mortar-based mechanical contact problems.

System Simplification

To illustrate the saddle point structure of the Jacobian matrix and how it can be simplified by static condensation of the Lagrange multiplier, we take the Jacobian matrix from a simple diffusion problem with equal value constraint as an example. The system of equations at a typical time step can be written as a block matrix as follows, (1) Here, we omit the identifier of time for simplicity. The first subscript and denote the primary and secondary subdomains, respectively. The second subscript denotes the part of the subdomain that is in contact and denotes the rest of the subdomain. The blocks denote the respective stiffness matrices. The block represents the coupling between the Lagrange multipliers and the primal variable in the secondary subdomain. The block denotes the coupling between the Lagrange multipliers and the primal variable in the primary subdomain.

The system of equations from mortar-based mechanical contact has a similar saddle point character, but is more complex than Eq. (1), considering the primary and secondary surfaces being partially in contact. For readers who are interested in the Jacobian matrix from mortar-based mechanical contact, please refer to (Popp et al., 2010).

[Variables]
 [./lm]
   order = FIRST
   family = LAGRANGE
   use_dual = true
 [../]
[]

[Contact]
  [frictionless]
    primary = plank_right
    secondary = block_left
    formulation = mortar
    c_normal = 1e0
  []
[]

VCP Workflow

A schematic is included in Figure 1 to demonstrate the solution process of VCP. Compared to the standard precondition and solve procedure, VCP features two additional customized computation steps (i.e., one to condense out the variable and the other to obtain the full solution vector) (see Figure 1). During static condensation of the variable (e.g., ), the DoFs are obtained for both the variable itself () and its coupled variable (). Based on this information, the coupling matrices (i.e., and ) will be extracted from the original Jacobian matrix. Then a reduced system of equations will be obtained with the necessary submatrix operations following Eq. (3). After solving the reduced system of equations, we obtain the primal variable solution vector, which is then utilized to update the variable (see Eq. (2)) and assemble the full solution vector.

Figure 1: The variable condensation preconditioner (VCP) workflow. Blue represents the original preconditioning steps. Orange shows the additional steps customized for VCP.

Performance

As an illustration, we solve the 2D Hertzian problem by using VCP with several sub-preconditioner types, including BoomerAMG (AMG), successive over-relaxation (SOR), additive Schwarz method (ASM), and block Jacobi (BJAC). For all the cases, we set adaptive_condensation = true and is_lm_coupling_diagonal = true. Note that here, ASM, and BJAC converge for the mortar-based mechanical contact problem, both as a standalone preconditioner and as a sub-preconditioner. AMG and SOR stagnate as a standalone preconditioner, while converge well as a sub-preconditioner under VCP. Figure 2 shows the performance of VCP in terms of total wall time and total number of linear iterations. It can be seen that VCP can pretty efficiently converge using AMG and SOR, whereas the standard preconditioner stagnates or diverges. For the other sub-preconditioner types, VCP is still more efficient than the standard preconditioner, despite the additional computational cost from the variable condensation step. This is because a better-conditioned system is obtained after static condensation of the Lagrange multipliers. For readers who are interested in getting more details, please refer to (Yushu et al., 2021). Note the performance of VCP for problems with increased complexities, e.g., larger, multi-physics problems, is to be examined in the future.

Figure 2: Performance of VCP for the Hertzian problem. Different sub-preconditioner types are utilized for comparison. Results are shown for four processors. Note the “inf.” identifies the case when the solver stagnates thus the wall time and total number of iterations go to infinity.

Example Input File Syntax

[Preconditioning]
  [vcp]
    type = VCP
    full = true
    lm_variable = 'leftright_normal_lm'
    primary_variable = 'disp_x'
    preconditioner = 'AMG'
    is_lm_coupling_diagonal = true
    adaptive_condensation = true
  []
[]

Input Parameters

References