Equation Solving Methods for Nonlinear Calculations

Technical Article on the Topic Structural Analysis Using Dlubal Software

Technical Article

This article explains the equation solver for a nonlinear calculation with a Newton-Raphson iteration.

Introduction

The finite element method is used whenever mechanical problems cannot be solved analytically. Often, nonlinear effects such as failure under pressure (geometric nonlinearity), plasticizations (material nonlinearity), and contact or kinematic degrees of freedom are also taken into account. These effects, especially for nonlinear material models, can be considered by using an iterative calculation method.

FEM Setting

Basic steps for FEM setting (further information can be found in [1]):

1. Weak form of equilibrium

Weak Form

$$∫Bgradδu⋯σdV⏟nonlinear component=∫Bδu·ρbdV+∫∂Bδu·todA$$

 δu Virtual (test) displacement t0 Initial load factor σdV Internal forces ρbdV Solid forces B Integrated area
2. Transformation in Voigt Notation with 4th order tensor

Weak Form in Voigt Notation

$$∫Bδε⋯ℂ⋯εdV = ∫BδuTbdV⏟solid forces + ∫∂BδuT · t0dA⏟surface area forces$$

 C Stiffness matrix δε Variation of the strain state B Integrated area ε Strain U Deformation

This notation is used in the following to solve the approximate solution of the FE approach for nonlinear materials.
3. For this, the displacement field is multiplied by the approach functions.

Displacement Field via Trial Function

$$u(x,t) = H(x)u^(t)$$

 u(x,t) Displacement over time (load increment) H Form function û Nodal displacement
4. Inserting derivative of displacement into the weak form. Numerical integration is used to calculate the nodal displacement and stresses and strains in post-processing by means of the material rule.

Sequence of Newton-Raphson Iteration

Due to the nonlinear material behavior, the material matrix C in Equation 2 above changes with each expansion step. The standard calculation method for solving this problem is the so-called Newton-Raphson iteration. It is used to linearize the function in a starting point. In the iteration, the stiffness matrix C of the preliminary step is always used. In the linearized iteration step, a tangent is placed at the zero of the function.

The equations belonging to the flowchart in the figure above are the following:

1. Dividing the load into load steps.

$$fextt + ∆t = fextt + ∆f$$

2. Predictor Step

Predictor Step

$$K0t∆0φ = fextt+∆t - fint0t$$

 K Stiffness matrix of the previous time step t+Δtfext External forces increased by a further load step 0tfint Internal forces of the previous time step ϕ Strain
3. In point 3, iteration of the flowchart, the total distortion reduced by the plastic distortion is calculated (corrector step).

The objective of the iterative calculation is always that the sum of the loads is zero. However, this is not possible numerically. Therefore, a break-off limit ε is defined at which the calculation is interrupted as sufficiently accurate.

Break-Off Limit

$$R = fextt + ∆t - fintnt + ∆t < ε$$

 R Break-off limit fext External forces fint Internal forces ε Epsilon break-off limit t Time step

In the program, the break-off limit can be set among the calculation parameters.

The following figure shows the flow of a Newton-Raphson iteration. In the first iteration

1. Iteration

$$Rt + ∆t - F0t + ∆t$$

 R Break-off limit t Time step F Force

the break-off R or ε is not reached. The tolerance limit is not reached in the second iteration (red) either. Only in the third iteration, the distance of the tangent stiffness is so small that convergence is achieved.

As already mentioned, the deformation is continuously summed up during the iteration.

Summary

The Newton-Raphson iteration has the consistency or convergence order 2. The number of "correct" locations in the iteration doubles with each step. Thus, a Newton-Raphson iteration converges quadratically, and the accuracy increases with each iteration when the method converges. However, if the method does not converge, the error goes to infinity and the calculation is stopped.

Error causes are, for example, a slope of the load-deformation curve that is too steep, and a curve's gradient in the plastic area that is too flat. If the load-deformation curve in the figure above showed a too strong break in the second iteration step, the material tangent and thus the stiffness matrix would not correctly display the slope of the elastic area. In this case, the slope of the root would be wrong for the plastic area. This is one of the reasons why an increase in the load steps is accompanied by an improved convergence.

Dipl.-Ing. (FH) Bastian Kuhn, M.Sc.

Product Engineering & Customer Support

Mr. Kuhn is responsible for the development of products for timber structures and provides technical support for our customers.

Reference

 [1] Nackenhorst, U.: Vorlesungsskript Numerische Mechanik. Hannover: Institut für Baumechanik und Numerische Mechanik, Gottfried Wilhelm Leibniz Universität, 2013

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