RD-E: 1602 IMPLICIT Solver

A dummy is sat down via gravity using the implicit approach (static).

The main advantages of implicit resolution are:
  • Unconditional stable scheme
  • Large time step
  • Treatment of the static problem

Options and Keywords Used

Input Files

Before you begin, copy the file(s) used in this example to your working directory.

Linear and Nonlinear Analysis

However, the implicit algorithm uses a global resolution which requires convergence for each time step and has low robustness in comparison to the explicit (null pivots, divergence for high nonlinearities, etc.).

The implicit methods result in solving a linear system for each time step, which is relatively expensive but enables a large time step: few expensive calculations. The explicit method treats linear or nonlinear systems depending on the problem. It is less expensive and faster for each step, but requires short time steps to ensure stability of the scheme that has many inexpensive cycles.
Newmark β
Implicit integration scheme

This scheme is unconditionally stable, the stability condition being independent of the time step choice. See the Radioss Theory Manual for further information about the Newmark scheme.

Radioss has a linear and a nonlinear solver. Only static computations are available and loading should be defined as a monotonous increasing time function for nonlinear analysis.

The main computational methods available in Radioss:
  • Cholesky (direct method, linear solver)
  • Preconditioned Conjugate Gradient (linear solver)
  • Modified Newton-Raphson method (nonlinear solver)
The precondition methods for linear solver available in Radioss:
  • No preconditioned
  • Diagonal Jacobi
  • Incomplete Cholesky
  • Stabilized incomplete Cholesky
  • Factored Approximate Inverse (by default)

You should define the tolerance and stop criterion for the linear and nonlinear solver (residual).

Strategies of resolution for nonlinear static computation/time step control:
  • Iterations number limit for updating stiffness matrix
  • Convergence iterations number for increasing time step
  • Convergence iterations number for decreasing time step
  • Increase time step factor
  • Decrease time step factor
  • Minimum time step
  • Maximum time step
  • Initial time step
The nonlinear solver uses the modified Newton-Raphson method and the resolution is based on sparse iterative techniques.

rad_ex_fig_16-19
Figure 1. Newton-Raphson Resolution in the Case of Load Control Technique

The modified Newton-Raphson method is based on maintaining the tangent matrix for all iterations and can be combined with the line search acceleration technique for accelerating convergence.

Piloting techniques available in Radioss:
  • Displacement norm control
  • Arc-length control

An automatic time step control is used.

Static Analysis and Implicit Options

This example deals with two implicit analysis':
  • A static linear computation (loading by gravity)
  • A static nonlinear computation (three computations are performed: dummy positioning using an imposed displacement, followed by a concentrated load and a gravity loading).

An adapted modeling methodology is set up for each analysis. Contact with the different interfaces depends on the computations taken into account and then the material can be updated.

The goal for this analysis is to propose a modeling method for different loading cases, with specific input data used in the implicit strategies. The studies by linear implicit and nonlinear implicit using imposed displacement are no longer comparable with results obtained by explicit, due to the different physical approaches. Comparisons are only valid for the positioning by gravity loading.

Model Description

Linear Static Analysis

TYPE7 interface uses nonlinear algorithms to check contact. Thus, in order to be used in a linear solver, it is replaced by a TYPE2 tied interface which creates kinematic conditions between secondary nodes and main surfaces. Gravity loading is studied.

rad_ex_fig_16-20
Figure 2. TYPE2 Tied Interface Linear Contact for Dummy/Seat Cushion Modeling
The visco-elastic LAW35 (generalized Kelvin-Voigt model) describing the foam of a seat is converted into a linear plastic LAW1 (properties are maintained):
Material Properties
Young's modulus
0.2 [ MPa ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaai Gac2eacaGGqbGaaiyyaaGaay5waiaaw2faaaaa@3BE6@
Poisson's ratio
0
Density
4.3 x 10-11 k g/l

Model Method

You can select BATOZ formulations for the shell elements and HA8 formulations using 2x2x2 integration points for the brick elements.

The linear implicit methods used are:
Implicit type
Static linear
Linear solver
Direct Cholesky
Precondition method
Factored Approximate Inverse
Stop criteria
Relative residual of preconditioned matrix
Tolerance
10-6
The implicit options used in the Engine file are:
/IMPL/PRINT/LINEAR/-1
Printout frequency for linear iteration
/IMPL/SOLVER/3
Solver method
5 0 3 0.0
/IMPL/LINEAR
Static linear computation

Results

Only one animation corresponds to the static solution.

rad_ex_fig_16-21
Figure 3. Linear Static Implicit Solution of Gravity Loading (TYPE2 interface is used)
It should be noted that this modeling contact slightly modifies the problem, which is no longer comparable with the previous explicit models.
Table 1. Indication of Time Computation
  Explicit Solver - /DYREL Implicit Solver - Linear
Normalized CPU 170 1

Nonlinear Static Analysis

Positioning Using an Imposed Displacement

The modeling methodology defined in the explicit studies is maintained (visco-elastic material law, TYPE7 interface, etc.). Brick elements are modeled by default element formulation.

rad_ex_fig_16-22
Figure 4. Nonlinear Contact Modeling with Self-impacting TYPE7 Interface
In addition to the constant gravity load, an imposed displacement along the Z-axis is applied on the main node of the global rigid body covering the dummy. This approach allows computation to converge and the rigid body modes to be removed (no null pivot). An input curve for the imposed displacement is required. The boundary conditions on main node 14199 are: 110 111.

rad_ex_fig_16-23
Figure 5. Imposed Displacement Along the Z-axis as a Monotonous Increasing Time Function
The nonlinear implicit parameters used are:
Implicit type
Static nonlinear
Nonlinear solver
Modified Newton
Stop criteria
Relative residual in force
Tolerance
0.01
Update of stiffness matrix
5 iterations maximum
Time step control method
Arc-length method and "line-search"
Initial time step
5 s
Minimum time step
0.01 s
Maximum time step
no
Desired convergence iteration number
6
Maximum convergence iteration number
20
Decreasing time step factor
0.8
Maximum increasing time step scale factor
1.1
Arc-length
Automatic computation
Spring-back option
no

Implicit parameters are set in the Engine file with the options beginning with /IMPL.

The implicit options used are:
/IMPL/PRINT/NONLIN/-1
Printout frequency for nonlinear iteration
/IMPL/SOLVER/3 5 0 3 0.0
Solver method (solve Ax=b)
/IMPL/NONLIN 5 2 0.01
Static nonlinear computation
/IMPL/DTINI 1
Initial time step determines the initial loading increment
/IMPL/DT/STOP 1e-3 0
Min-Max values for time step
/IMPL/DT/2 6.0 20 0.8 1.1
Time step control metho 2 - Arc-length+Line-search will be used with this method to accelerate and control convergence

Due to the contact problem, the tolerance value (Tol) is set to 10-2 (default = 10-3).

Some options are not compatible with the implicit solver. Refer to Radioss Starter Input for more details about implicit options.

Results

The last animation corresponds to the static solution.


rad_ex_fig_16-24
Figure 6. Nonlinear Static Implicit Solution of the Imposed Displacement
The Z-displacement of the dummy should not be considered as a result but as an input data (imposed displacement on the main node 14199).
Table 2. Time Computation Comparison between Explicit and Implicit Computations
  Explicit Solver - /DYREL Implicit Solver - Nonlinear
Normalized CPU 1.26 1
Number of cycles (normalized) 56704 (1718) 33 (1)

Positioning Using a Concentrated Load

The modeling methodology defined in the explicit studies is maintained. The gravity loading is taken into account by applying a constant concentrated load of 813.05N (dummy weight + added masses) on the main node of the rigid body, including the dummy. In order to remove the rigid body modes, the dummy is connected to fixed nodes via TYPE8 spring elements.

rad_ex_fig_16-25
Figure 7. Concentrated Load Along the Z-axis as a Monotonous Increasing Time Function

rad_ex_fig_16-26
Figure 8. Springs TYPE8 Defined for Removing Rigid Body Modes during Implicit Computation
The properties of the general TYPE8 springs are:
  • Linear elastic behavior
  • Mass = 1g
  • Inertia = 0.001
  • Translational stiffness: TX = 1 N/mm

    TY = 1 N/mm

    TZ = 1 N/mm

  • Rotational stiffness: RX = 100 Mg.mm2/(s2.rad)

    RY = 100 Mg.mm2/(s2.rad)

    RZ = 100 Mg.mm2/(s2.rad)

Implicit options are the same as the previous implicit problem; except for the initial time step is set to: 2s.

Results

Table 3. Time Computation Comparison between Explicit and Implicit Computations
  Explicit Solver - /DYREL Implicit Solver - Nonlinear
Normalized CPU 3.07 1
Number of cycles (normalized) 56704 (1090) 52 (1)
Z - displacement (main node dummy) -12.75 mm -12.49 mm

Positioning Using Gravity Loading

The modeling methodology defined in the implicit model is maintained using a concentrated load. Gravity loading is applied on the secondary nodes and the main node of the rigid body, including the dummy. In order to remove the rigid body modes, the dummy is connected to fixed nodes via TYPE8 spring elements.

rad_ex_fig_16-27
Figure 9. Gravity Loading as a Monotonous Increasing Time Function

Implicit options are the same as the previous implicit problem (initial time step is set to: 2s).

Results

Table 4. Time Computation Comparison between Explicit and Implicit Computations
  Explicit solver - /DYREL Implicit solver - Nonlinear
Normalized CPU 2.53 1
Number of cycles (normalized) 56704 (1090) 52 (1)
Z - displacement (main node dummy) -12.75 mm -12.42 mm

rad_ex_fig_16-28
Figure 10. Convergence results of the X- and Z-displacement of main node 14199 (rigid body dummy). for the implicit models using gravity loading and concentrated load

rad_ex_fig_16-29
Figure 11. Final dummy position obtained using IMPLICIT (model using gravity loading) . (and EXPLICIT model with gravity loading and kinetic relaxation)

Conclusion

This example brings awareness to the use of the Radioss implicit solver in resolving quasi-static problems. On the other hand, it illustrates different convergence acceleration techniques when an explicit solver is applied to the quasi-static problems. The advantages and drawbacks of the methods are compared.