Equation Solving Methods for Nonlinear Calculations
Technical Article
This article explains the equation solver for a nonlinear calculation with a NewtonRaphson 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]):
 Weak form of equilibrium
${\int}_{B}\underset{\mathrm{nonlinear}\mathrm{component}}{\underset{\u23df}{grad{\delta}_{u}\cdots \sigma dV}}={\int}_{B}{\delta}_{u}\xb7\rho bdV+{\int}_{\partial B}{\delta}_{u}\xb7{t}_{o}dA$
δ_{u} Virtual (test) displacement t_{0} Initial load factor σdV Internal forces ρbdV Solid forces B Integrated area  Transformation in Voigt Notation with 4^{th} order tensor
${\int}_{\mathrm{B}}\mathrm{\delta \epsilon}\cdots \mathrm{\u2102}\cdots \mathrm{\epsilon dV}={\int}_{\mathrm{B}}\underset{\mathrm{solid}\mathrm{forces}}{\underset{\u23df}{{\mathrm{\delta u}}^{\mathrm{T}}\mathrm{bdV}}}+{\int}_{\partial \mathrm{B}}\underset{\mathrm{surface}\mathrm{area}\mathrm{forces}}{\underset{\u23df}{{\mathrm{\delta u}}^{\mathrm{T}}\xb7{\mathrm{t}}_{0}\mathrm{dA}}}$
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.  For this, the displacement field is multiplied by the approach functions.
$\mathrm{u}(\mathrm{x},\mathrm{t})=\mathrm{H}\left(\mathrm{x}\right)\hat{\mathrm{u}}\left(\mathrm{t}\right)$
u(x,t) Displacement over time (load increment) H Form function û Nodal displacement  Inserting derivative of displacement into the weak form. Numerical integration is used to calculate the nodal displacement and stresses and strains in postprocessing by means of the material rule.
Sequence of NewtonRaphson 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 socalled NewtonRaphson 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:
 Dividing the load into load steps.
${}^{\mathrm{t}+\u2206\mathrm{t}}\mathrm{f}_{\mathrm{ext}}={}^{\mathrm{t}}\mathrm{f}_{\mathrm{ext}}+\u2206\mathrm{f}$
 Predictor Step
${}_{0}{}^{\mathrm{t}}\mathrm{K}{\u2206}_{0}\mathrm{\phi}={}^{\mathrm{t}+\u2206\mathrm{t}}\mathrm{f}_{\mathrm{ext}}{}_{0}{}^{\mathrm{t}}\mathrm{f}_{\mathrm{int}}$
K Stiffness matrix of the previous time step ^{t+}Δ^{t}f_{ext} External forces increased by a further load step _{0}^{t}f_{int}
Internal forces of the previous time step ϕ Strain  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 breakoff limit ε is defined at which the calculation is interrupted as sufficiently accurate.
$\mathrm{R}=\left{}^{\mathrm{t}+\u2206\mathrm{t}}\mathrm{f}_{\mathrm{ext}}{}_{\mathrm{n}}{}^{\mathrm{t}+\u2206\mathrm{t}}\mathrm{f}_{\mathrm{int}}\right\mathrm{\epsilon}$
R  Breakoff limit 
f_{ext}  External forces 
f_{int}  Internal forces 
ε  Epsilon breakoff limit 
t  Time step 
In the program, the breakoff limit can be set among the calculation parameters.
The following figure shows the flow of a NewtonRaphson iteration. In the first iteration
${}^{\mathrm{t}+\u2206\mathrm{t}}\mathrm{R}{}^{\mathrm{t}+\u2206\mathrm{t}}\mathrm{F}^{0}$
R  Breakoff limit 
t  Time step 
F  Force 
the breakoff 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 NewtonRaphson iteration has the consistency or convergence order 2. The number of "correct" locations in the iteration doubles with each step. Thus, a NewtonRaphson 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 loaddeformation curve that is too steep, and a curve's gradient in the plastic area that is too flat. If the loaddeformation 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.
Author
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.
Keywords
Convergence Load step Time step
Reference
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