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The Finite Element method ( FEM ), originated for structural analysis. Its root goes to the earlier work of Courant in The main Ideas of FEM has the following three features :- 1- Divide the whole domain into parts, called finite elements. 2- Over each element, relations will developed among the secondary and primary variables; ( force and displacements, heat and temperature,..etc). 3- then, assembling the elements ( which means combine the relations of all elements) to obtain the relations between the secondary and primary variables of the system.

The main characteristics of FEM Analysis are :- 1- Continuum mechanics 2- Finite element discretization 3- Numerical algorithms 4- Software considerations.

Finite element Problem FieldPrimary Variable Problem DATASecondary variable uacfQ Elastic barsDisplacementAxial stiffness (EA) 0Axial force (f)Point force (p) Heat transfereeTemperature (T-T ∞ ) Thermal conductance (kA) Surface convection (Apβ) Heat generation (f) Heat (Q) Pipe flowPressure (P)Flow resistance (1/R) 00Point force (Q)

FieldPrimary Variable Problem DATA Secondary variable uacfQ Elastic barsDisplacementAxial stiffness (EA) 0Axial force (f)Point force (p) Heat transfereeTemperature (T-T ∞ ) Thermal conductance (kA) Surface convection (Apβ) Heat generation (f) Heat (Q) Pipe flowPressure (P)Flow resistance (1/R) 00Point force (Q)

Third: Interpolation function To complete the solution, the smoothness of the approximated solution should be satisfied also. Interpolation of the function develop differentiable solution. The Interpolation form called also the shape function

Nonlinear Finite Element

The nature of the problem makes the invariant of coefficient” stiffness” matrix logically impossible. One way of getting solution is the Iterative methods. It based on assuming solution vector and start controlled iteration with predicted New solutions

Shape memory materials Alloy or polymers; shape memory materials “ remember” its original shape. Shape memory alloy – NITINOL will be taken as example. Shape memory effect happened due the materials microstructure in which there exists two different crystallography structures, which are; austenitic and martensitic phases.

Tem. Stress

Wikipedia

ANSYS Inc. Documentation

Workbench APDL

Summery Linear Fem sets should have a unique solution, while nonlinear fem set may Not Iterative methods used on Nonlinear FEM and not Always converge or sometimes, may converge to the wrong solution. The solution to nonlinear FEM is sensitive to Initial and boundary conditions due to coefficient dependency.

Special thanks to ANSYS for their advices; and valuable free software That, they provided.