Equilibrium Method

Activating the command: Setup Model Setup Parameters Advanced Equilibrium tab

The information on this page applies to the following analysis types except if indicated:

Mechanical Event Simulation (MES)

Static Stress with Nonlinear Material Models

In nonlinear finite element analysis, most iterative solution schemes are based on some form of the well-known Newton-Raphson iteration algorithm. A detailed description of the Newton-Raphson scheme may be found in many references, including Hinton, Oden and Stricklin. The user is also encouraged to review Section 11.5 (pp. 449-452) of Linear and Nonlinear Finite Element Analysis in Engineering Practice by Constantine C. Spyrakos and John Raftoyiannis for more discussion on iterative solution schemes.

There are a number of nonlinear iterative solution methods available. (Performing a Riks analysis sets the iterative solution method.) All the methods are based upon the Newton-Raphson iteration scheme. You choose from the following methods that display in the Nonlinear iterative solution method drop-down box:

These methods are discussed in more detail below.

Combined Newton (Combined Full / Modified Newton-Raphson Method)

The combined full and modified Newton-Raphson method is between the full Newton-Raphson method and the modified Newton-Raphson method, and is designed for users who either have some prior knowledge of the structures at hand or have some advanced knowledge on nonlinear structural behaviors. You can specify a particular iterative scheme that may best suit your problem. The full Newton-Raphson and the modified Newton-Raphson methods are special cases of this method. The default scheme for the combined full-modified Newton-Raphson method is two right-hand side effective load vector updates for each effective stiffness matrix reformation.

The latest solution method allows the analysis to achieve convergent solutions for problems involving motion. This solution method damps out common convergence problems such as high frequencies, because they are just noise within the solution.

Full Newton 1 (Full Newton-Raphson Method)

The full Newton-Raphson iterative solution scheme, or the tangent stiffness matrix method, is the basic form of all the schemes. In this solution scheme the effective stiffness matrix and the right-hand side effective load vector of the system are reformed or updated for each equilibrium iteration within all the time/load steps. The advantages of this method are that it is usually more effective for problems with strong nonlinearity, and that it converges quadratically with respect to the number of iterations. Since in general the major cost per equilibrium iteration for nonlinear analysis lies in the construction and factorization of the effective tangent stiffness matrix, the full Newton-Raphson scheme may be more expensive in turn of solution time, especially for large-scale problems.

Full Newton 2

Full Newton 2 is a hybrid of the Full Newton 1 and BFGS solution methods.

The BFGS method is one kind of quasi-Newton solution method, as presented in the references listed below.

When the Full Newton 2 option is selected, the BFGS method is generally used. The following two cases are the exception, for which the Full Newton 1 method is used instead:

  • For elasto-plastic material models when the updated Lagrangian formulation is specified (for 2D, brick, and tetrahedron elements)
  • For elasto-plastic material models when the incompatible mode is enforced (for brick elements)

References:

  • Matthies, H. and Strang, G. – The Solution of Nonlinear Finite Element Equations

    International Journal for Numerical Methods in Engineering

    Vol. 14, 1613-1626 (1979)

  • Lee, S. – Rudimentary Considerations for Effective Quasi-Newton Updates in Nonlinear Finite Element Analysis

    Computers & Structures

    Vol. 33, n2, 463-476 (1989)

  • Lee, S. – Rudimentary Considerations for Effective Line Search Method in Nonlinear Finite Element Analysis

    Computers & Structures

    Vol. 32, n6, 1287-1301 (1989)

Modified Newton (Modified Newton-Raphson Method)

The modified Newton-Raphson iterative solution scheme is a procedure that lies in between the tangent stiffness matrix method (the full Newton-Raphson method) that reforms the effective stiffness matrix for each equilibrium iteration within all the time/load steps, and the initial stiffness matrix method (the initial stress method) that constructs and factorizes the effective stiffness matrix only once. The modified Newton-Raphson method performs the reformation of the effective stiffness matrix only for the first equilibrium iteration within each time step, and the rest of the iterations will only involve the updating of the right-hand side effective load vectors.

Since the modified Newton-Raphson method involves fewer effective stiffness matrix reformations and factorizations, the computational cost per iteration for the modified Newton-Raphson method is usually much less than that for the full Newton-Raphson method. It has been observed that for problems with mild or moderate nonlinearity, for example, smooth material property or loading condition changes, the modified Newton-Raphson method is usually more effective. However, for problems with strong nonlinearity, for example, sudden material property or loading condition changes, this method may converge very slowly or even diverge.

Line Searches

All of the solution schemes have the option for including line searches. With the exception of Full Newton 2, the line search can be excluded from the available solution methods.

Line searching usually helps to stabilize the iterative schemes. It can be particularly useful for problems involving rapid changes in structural stiffness due to rapid material property and/or geometric configuration changes. In such situations line searching usually can accelerate the iterative process, and sometimes provide convergence where none is obtainable without line searches. The basic idea behind a line search scheme is the following: During each equilibrium iterative, the Newton-Raphson method generates a search direction for new possible solutions, while the line search scheme is used to find a solution in that direction that minimizes the out-of-balance force error. The convergence tolerance for the line search can be specified in the Line search convergence tolerance field. This value should be between 0.4 and 0.6.

Selection of Iterative Solution Scheme

Unlike linear problems, in nonlinear analysis there is no solution scheme that is good for all kinds of problems. The choice of iterative scheme may be more appropriately chosen for a particular problem based on the degree of nonlinearity of the problem at hand. Problems with strong material and geometric nonlinear responses usually require more frequent matrix reformations. The modified Newton-Raphson method is usually more effective for problems with smooth material property and/or geometrical configuration changes, while the full Newton-Raphson method, although more expensive in terms of numerical cost per iteration, is usually more effective than the modified Newton-Raphson method for problems of strong nonlinearity. Line searching schemes help an iterative process to converge at sensitive time/load levels but increase the computational cost per iteration. For a nonlinear analysis where no prior knowledge is available on the behavior of the structure at hand, the following procedures are recommended:

  1. Start the analysis with the material nonlinear only analysis type and a linear material model, such as constant material properties. If the material model available for the analysis is nonlinear, then use the initial values derived from the material model/curves as the constant material properties. The linear analysis results may not only help to check whether the geometric, loading and boundary conditions of the system are properly set or imposed, but may also provide some useful information on the initial response or behavior of the structure since for small displacements all structures behave linearly. Displacement results from the linear analysis may also provide a criteria for the selection of reasonable time/load step increments for the on-going nonlinear analysis.
  2. If the original material model is nonlinear, perform the analysis using the material nonlinear only analysis type. Results from this analysis may help to further understand the behavior of the structure response and identify some sensitive time/load levels. Moreover, the comparison of the results from this analysis and those from a large displacement analysis will indicate whether or not the object has entered a large displacement state. For a structure of small displacement state, the results from all the three analysis types should be more or less the same. The time/load level where the results from the three analysis types starts to depart from each other significantly indicates that the structure has entered large displacement state.
  3. With the knowledge obtained from (1) and (2), one may perform the analysis with the Total Lagrangian or Updated Lagrangian analysis types. If the automatic time step option is used, the time/load step increments must be reasonably small to represent the material property and/or geometrical configuration changes of the object during its loading process. If the multiple time step option is used, one may assign different time/load steps and other parameters for different time/load zones. For those time/load zones where rapid material property and geometrical changes may occur, smaller time steps, more frequent matrix reformations, and line searches are usually required. At some critical time/load levels, such as bifurcation or collapse time/load levels, it may be necessary to avoid performing matrix reformations and/or equilibrium iterations or to relax error tolerances near these time/load levels to observe the pre- or post-buckling/collapse behaviors of the object.

Parent topic: Advanced Settings