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Nonlinear Iteration and Time Integration

FrontISTR advances the solution by combining nonlinear iterations and time integration at each substep within the increment control of each step. This page describes the iterative structures located between the outer step control and the inner linear solver: Newton-Raphson iterations, contact iterations, time integration for dynamic analysis, and the framework for time advancement and nonlinear iteration in transient heat conduction.

Overview

Nonlinear solution and time integration within a step have a nested structure consisting of multiple loops from the outside inward. Each layer has the following role.

Layer Role Main reference page
Step loop Divides the entire analysis into multiple steps and switches boundary conditions, loads, contact, and element activation. Step Control
Substep loop Divides one step into time increments and advances using fixed increments or automatic increments and cutback. Step Control
Time-integration state advancement In dynamic analysis and transient heat conduction, obtains the state at the next time from the state of the previous step. This page
Contact-state search loop In contact analysis, searches for changes in contact state (contact/separation) and updates the structure of the stiffness matrix. This page
Augmented Lagrange outer iteration With ALAGRANGE contact, progressively improves contact-constraint accuracy. This page
Newton iteration Iteratively solves nonlinear equilibrium equations using the Newton-Raphson method. This page
Linear-system solution Solves the linear system based on the tangent stiffness matrix at each Newton iteration. Solver and Preconditioning

Which of these layers appear depends on the analysis type.

Analysis type Time integration Contact-state search Outer iteration Newton iteration
Linear static analysis — — — —
Nonlinear static analysis (without contact) — — — Yes
Nonlinear static analysis (SLAGRANGE contact) — Yes — Yes
Nonlinear static analysis (ALAGRANGE contact) — Yes Yes Yes
Dynamic analysis, implicit Newmark-β — — Yes (when nonlinear)
Dynamic analysis, explicit Central difference — — —
Frequency response analysis Modal superposition — — —
Transient heat conduction Crank-Nicolson / Backward Euler — — Yes (when material properties depend on temperature)
Steady heat conduction — — — Yes (when material properties depend on temperature)

For how to select the contact type and algorithm themselves, see Contact and Embedding. This page describes how the selected algorithm is organized as an iterative loop.

Choosing the Solution Scheme

The iterative structure is determined automatically from the selected analysis type (Analysis Types), the presence or absence of nonlinearity (geometric nonlinearity, material nonlinearity, or contact), and the contact algorithm (SLAGRANGE or ALAGRANGE). The user directly adjusts the upper limits on iteration counts and the convergence criteria.

The following guidelines apply when tuning these settings.

  • For standard analyses, the default upper limit on the number of iterations is generally sufficient. If a difficult-to-converge problem reaches the maximum number of iterations without convergence, first use cutback (Step Control); increase the upper limit only if this does not improve convergence.
  • Adjust convergence criteria to balance analysis stability and computation time. Stricter criteria improve equilibrium accuracy but increase the number of iterations. In contact analyses and analyses involving Lagrange multipliers, combining displacement-increment or Lagrange criteria with the residual criterion can improve stability.
  • For the time-integration parameters in dynamic analysis (the Newmark-β coefficients), start with a standard combination of values and adjust them when changing the numerical-damping characteristics of the vibration response.
  • For the time-integration scheme in transient heat conduction, use Crank-Nicolson for problems with smooth temperature changes, and Backward Euler when sharp changes or long-term stability are more important.

Newton-Raphson Iteration in Static Analysis

In nonlinear static analysis, equilibrium equations containing geometric or material nonlinearity are solved iteratively by the Newton-Raphson method. Each iteration proceeds as follows.

  1. Evaluate the residual vector \(\boldsymbol{R}\) at the current solution.
  2. Assemble the tangent stiffness matrix \(\boldsymbol{K}_T\).
  3. Solve the linear system \(\boldsymbol{K}_T \Delta \boldsymbol{u} = -\boldsymbol{R}\) to obtain the displacement correction \(\Delta \boldsymbol{u}\).
  4. Update the solution as \(\boldsymbol{u} \leftarrow \boldsymbol{u} + \Delta \boldsymbol{u}\).
  5. Compare against the convergence criteria. If convergence is achieved, end the iteration; otherwise return to step 1.

The linear solver performs the linear-system solution (step 3). For selection of the solver and preconditioner, see Solver and Preconditioning.

Iteration is controlled using the step-control parameters of !STEP. The main control values are as follows.

Role Parameter
Upper limit on the number of iterations. If the limit is reached without convergence, the substep is subject to cutback. MAXITER
Convergence criterion based on the relative residual norm. CONVERG
Convergence criterion based on the norm ratio of the displacement correction. Used together with the residual criterion to improve stability. CONVERG_DDISP
Divergence criterion for the residual norm. Iteration is terminated when this value is exceeded. MAXRES

The maximum Newton iteration count referenced by the automatic-increment and cutback decisions in Step Control means the number of Newton iterations described on this page. Automatic increment control increases the time increment after a substep that converged in few iterations and decreases it after a substep that required many iterations.

Contact Iteration in Static Analysis

In contact analysis, the contact state changes during iteration because of contact, separation, and sliding at contact surfaces, so Newton iteration is combined with contact-state updates. The structure depends on the contact-solution algorithm.

Both algorithms place a contact-state search loop at the outermost level. At each pass through this loop, the contact state (contact/separation) is reevaluated after the inner Newton iteration has converged. If the state has changed, the structure of the stiffness matrix is updated and the inner iteration is run again. The loop ends when the contact state is unchanged from the previous pass and the contact convergence criteria (contact-force and Lagrange-multiplier criteria) are satisfied. The upper limit on the number of passes is specified by MAXCONTITER of !STEP.

With SLAGRANGE contact, contact degrees of freedom are incorporated into the linear system using the standard Lagrange multiplier method, giving a two-layer structure in which Newton iteration is placed directly inside the contact-state search loop. No augmented Lagrange iteration is involved.

With ALAGRANGE contact, an augmented Lagrange iteration is added inside the contact-state search loop, and Newton iteration is performed inside that, giving a three-layer structure. At each augmented Lagrange iteration, the contact constraint is reevaluated using a combination of the penalty term and Lagrange multiplier, progressively improving constraint accuracy. The upper limit on the number of augmented Lagrange iterations is specified by AUGITER of !CONTACT_ALGO.

The control values for contact iteration are as follows.

Role Specified in
Upper limit on the number of contact-state search-loop iterations. Used for both SLAGRANGE and ALAGRANGE. !STEP MAXCONTITER
Upper limit on the number of augmented Lagrange iterations. Used only for ALAGRANGE. !CONTACT_ALGO AUGITER
Convergence criterion for the correction to the Lagrange multiplier. Used together with the residual criterion in analyses with contact. !STEP CONVERG_LAG

The contact iteration count referenced by automatic-increment decisions in Step Control means the number of iterations described on this page, including contact-state updates. Analyses in which contact and separation occur frequently tend to require more contact iterations, which also affects automatic-increment and cutback decisions. For contact types, pair definitions, and algorithm selection themselves, see Contact and Embedding. For how to solve the linear system containing contact degrees of freedom, see Solver and Preconditioning.

Time Integration in Dynamic Analysis (Implicit Method)

The implicit method for dynamic analysis uses the Newmark-β method to relate displacement, velocity, and acceleration at the next time at each time step and combines these relations with the equation of motion to obtain the state at time \(t + \Delta t\). The stability restriction on the time step \(\Delta t\) is mild, allowing structural responses dominated by low-frequency components to be solved efficiently.

The following two time-integration parameters are used.

Parameter Role
\(\beta\) Coefficient related to time interpolation of displacement.
\(\gamma\) Coefficient related to time interpolation of velocity.

The combination \(\beta = 1/4\) and \(\gamma = 1/2\) is called the average acceleration method and is a standard choice that is unconditionally stable and has no numerical damping. To introduce numerical damping, choose \(\gamma\) greater than \(1/2\). For mathematical details of parameter combinations, stability, and error characteristics, see the theory manual.

When nonlinearity (geometric nonlinearity, material nonlinearity, or contact) is included, Newton iteration is performed within each time step; the linear system is solved within the iteration and the state is updated. The convergence criteria use CONVERG and related values shared with static analysis. In dynamic analysis with contact, the same iterative structure as Contact Iteration in Static Analysis is incorporated into each time step.

The time step in dynamic analysis is primarily controlled by the time conditions of !STEP. With fixed increments, !STEP specifies DTIME as the time step and ETIME as the step duration. The n_step and t_delta values of !DYNAMIC are treated as defaults when !STEP is omitted and for backward compatibility. In the nonlinear implicit method, !STEP with INC_TYPE=AUTO enables automatic increments and cutback, increasing or decreasing the time step according to the state of Newton and contact iterations. Whether fixed or automatic increments are used, set the time step with both convergence and required accuracy in mind.

Time Integration in Dynamic Analysis (Explicit Method)

The explicit method for dynamic analysis uses the central difference method to calculate the state at the next time directly from only the displacement, velocity, and acceleration information at the previous time. Because no simultaneous equations are solved, the computation cost per step is low. Even when nonlinearity is included, each time step is a single-stage update without Newton iteration.

The time step has an upper limit imposed by a stability condition based on the minimum natural period of the system (the CFL condition). A time step exceeding this limit causes the numerical solution to diverge, so large time steps like those possible with the implicit method cannot be used. The explicit method is advantageous for phenomena such as impact, wave propagation, and high-speed contact, where small time steps are inherently required.

When contact is included, the Forward Increment Lagrange method is used to impose contact constraints. Contact forces are evaluated in a manner consistent with the single-stage update of the explicit method.

Solution Method for Frequency Response Analysis

Frequency response analysis directly obtains the periodic steady-state response in the frequency domain using the modal superposition method. It evaluates the response while varying the excitation frequency and does not need to trace a time history through iterations. Neither nonlinear iterations nor time integration are performed.

Before frequency response analysis can be performed, modal analysis must be carried out for the same system and the required number of natural modes must be extracted. Only linear models are supported; the analysis cannot be performed when geometric or material nonlinearity is enabled. For the positioning of this analysis type, see Analysis Types.

Time Advancement and Nonlinear Iteration in Transient Heat Conduction

Transient heat conduction has its own time loop completed within !HEAT. The time step is controlled through a path separate from the structural-analysis !STEP and !AUTOINC_PARAM, and when material properties depend on temperature, nonlinear iteration is performed within each time step.

The time-integration scheme is selected with the \(\beta\) parameter of !HEAT.

\(\beta\) Scheme Characteristics
0.5 Crank-Nicolson method Second-order accuracy. Suitable for problems with smooth temperature changes.
1.0 Backward Euler method First-order accuracy. Unconditionally stable and provides stability for long-duration analyses and sharp temperature changes.

The time step is controlled adaptively by combining the initial time step, minimum time step, and maximum temperature change per step. If the temperature change exceeds DELTMX after calculation at a time step, the time step is reduced and the step is recalculated. The analysis is aborted if the time step falls below the minimum time step DTMIN.

When material properties depend on temperature, nonlinear iteration is performed within each time step. Iteration is controlled as follows.

Role Parameter
Upper limit on nonlinear iterations. !HEAT ITMAX
Convergence criterion. !HEAT EPS

Steady heat conduction does not use time stepping; only nonlinear iteration is performed when the material properties depend on temperature. Because time control for heat conduction analysis is independent of the structural-analysis !STEP, check the !HEAT settings when adjusting the time step.