Method of Finite Elements II
Consider a MEMS application (micro-electro-mechanical system) where a magnetically sensitive MEMS bar is fixed
is ε(x). A magnetic field applied in the system causes a uniformly distributed tension force in the bar. The stresses
at one end. The bar is constrained to move only in the longitudinal direction and thus the only nonzero strain
develop in the bar σ(x) satisfy the equilibrium equation σ,x + f = 01. The bar is made of an Isotropic Hookean
bulk modulus K, the shear modulus µ and the failure work φ (all are in units of force to area). In this assignment we
nonlinear material with softening, as proposed in reference . This material is characterized by 3 constants: the
will define unitless terms by normalizing all quantities in the problem by φ:
���� ���� ���� 2 ����
���� ∗ = , ���� ∗ = , ���� = � , ���� =
���� �������� ���� 3 ����
a, β are material parameters.
Under these assumptions the material law is given by:
���� ∗ = ���� ∗ (���� )���� ����ℎ������������ ���� (���� ) = [����2 + 2����(1 + �������� )]���� −(��������+��������
The strong form of the problem is given by
σ∗ + ���� ∗ = 0
⎪���� = ���� (����)����
(S) ε = u,x
⎨ u(0) = 0
⎩ σ (L) = 0
where ���� - axial displacement, ���� - axial strain, ���� ∗ - normalized axial stress and ���� ∗ - normalized distributed load. The
following parameters are given:
���� = 1 [��������], ���� = 580, ���� = 1.7 × 105
����,���� = ��������
(ε − ε 0 ) (and set ����0 = 0), obtain the first order
∂E (ε 0 )
1. Using Taylor expansion: ET (ε ) = E (ε 0 ) +
approximation �������� (���� ) to the material law such that ���� ∗ (����) ≈ �������� (����). Plot the stress-strain curves of the exact
and approximated material laws.
2. Write a finite element code with linear shape functions and Newton iterations to solve this problem with the
the nodal displacement ����, the strain ���� and stress ���� ∗ in every element. The code should be able to implement both
exact and approximated material laws (Use the code provided in HW1 as basis). The output of the code should be
full Newton and modified Newton methods (use an index so the user can choose which method) and print
information about the convergence of the methods. The source files should be submitted electronically so they can
���� ���� , for each element as mentioned in the class notes. This will be equal to
Note: For the application of the Newton Raphson method you will need to calculate the tangent stiffness matrix
T =K +
Which in matrix component notation is written as:
2 ∂K e
Tij = K e + � u
k=1 ∂uj k
Then, after forming the total stiffness and tangent stiffness matrices and applying the boundary conditions, the
incremental displacement will be calculated as:
∆u = −Ttot r = −Ttot (K tot u − Ftot )
(r is the residual as referred to in the class notes)
3. Increase ���� ∗ gradually (with small increments) until the code is not converging (the increments in ���� ∗ should be
consistent with ~10-4 increments in the strain ����). Run the code with incremental loading using the following
a. Full Newton - the tangent stiffness T is computed at every step
b. Modified Newton 1 - the tangent stiffness T is only computed at the beginning of every load increment.
c. Modified Newton 2 - the tangent stiffness T is only computed every 3 loading increments.
d. Modified Newton 3 - the tangent stiffness T is only computed once at the beginning of the analysis and is
not updated during the run.
Choose the numerical parameters such as the density of the mesh, Newton convergence tolerances, initial guess,
number of increments etc. in a "reasonable way". The analysis should be carried out for both material models (8
4. Obtain the final maximum force ���� ̅∗ from question 3, in which Newton's method will converge. Run the code
without incremental loading using the following methods:
a. Full Newton - the tangent stiffness T is computed at every step.
b. Modified Newton 4 - the tangent stiffness T is only computed every 3 iterations.
c. Modified Newton 5 - the tangent stiffness T is only computed once at the beginning of the analysis and is
not updated during the run.
The analysis should be carried out for both material models (6 numerical experiments).
5. Finally, write a report including the following points:
a. Stress-strain curves from question 1.
b. Finite element formulation of the problem.
c. Give a short explanation of the main subroutines used in your code.
d. table with all numerical parameters chosen.
e. full description of the convergence for each run (14 runs).
f. Choose one of the converging methods and plot ����(����), ����(����), ���� ∗ (����) across the length of the bar ����, for 3 load
levels: 3 ���� ∗ , 3 ���� ∗ , ���� ∗ and both material laws.
e. Give a brief overview of Mathematical (e.g: why the method won’t converge after ���� ∗ , etc) and Physical
conclusions (behavior of the bar/material).
���� ∗ has been reached? Try to code and implement such a method on the above problem.
6. Extra credit: Can you think of a method that can cope with the convergence issue of the Newton method after
. K. Y. Volokh, "Softening hyperelasticity for modeling material failure: Analysis of cavitation in hydrostatic
tension", Int. J. of Solids & Struct., 44 (14-15): 5043-5055, 2007.