Finite Element Modelling Methods

Following on from the consideration of linear and non-linear modelling strategies, the methodology of carrying out a static and/or dynamic analysis must be discussed. For a static analysis, the time dependent effects of damping and inertia do not have a significant effect on the response of the structure. The basis of a static analysis is,

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where {F} is the force vector, [K] is the stiffness matrix of the structure and {u} is the displacement vector. Even a nonlinear static analysis can be performed (including contact) assuming that the load is applied gradually to ensure an accurate solution is reached.

In reality, all systems behave dynamically under the action of a load or displacement, however when application of the load or displacement is slow enough, the systems is assumed to be loaded and respond statically as the inertial forces are neglected. The importance of considering both static and dynamic loading for implanted hip joints has been studied in [190, 191]. Both static and dynamic analysis is considered for both linear and non-linear systems; however some limitations in computational techniques may mean that simplifying assumptions are required. The basis of dynamic equilibrium includes inertia effects and is represented by the following equation,

E

L I u

M ..+ = (3.15)

where M, ݑሷ and I are the mass, acceleration and internal forces of the structure respectively and LE is the externally applied load, which is based on Newton’s second Law of motion (F =

ma). This equation forms the difference between a static and dynamic analysis where a dynamic analysis includes the inertial effects and internal forces. In terms of a simplistic undamped system such as a spring-mass set-up, for linear dynamics to be applicable then the system must be fully linear along with further criteria defined in the literature [192]. As undamped systems are not valid in reality the analysis should account for a damped influence as a form of energy dissipation. By including the damping term within the dynamic equilibrium equation the following equation is obtained:

E L u C Ku u M ..+ − . = (3.16)

where C and ݑሶ is the damping matrix and velocity of the structure respectively. Through performing a structural dynamic analysis the effects of time varied loading on a structure is determined. Another way to state (3.16) is as follows,

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( )

{

}

{ }

{ }

{ }

where the load vector {F} is a function of time t.

Both implicit and explicit methods were used for developing finite element models and carrying out analysis. An explanation of both implicit and explicit methods for engineering problem solving is provided by Harewood and McHugh [193]. For an implicit approach, a solution is obtained by iteration at each increment to obtain a solution to the finite element equations. An explicit solution is determined directly at each increment without iteration or a convergence criterion. Both implicit and explicit solvers are available in the finite element analysis software used in this project, and each solver will be discussed in the more detail below.

The implicit finite element solver determines the unknowns {u} and obtains the equilibrium at time t whilst including the effects of damping and inertia. The explicit method includes failure material models, complex contact and impact whilst using less memory and data storage space. The implicit analysis is used for statically and quasi-statically loaded models [194]. For more complex contact conditions such as the laxity based pure microseparation models, the explicit method provides a more efficient solution as the implicit method is required to solve a large number of linear equations. The Abaqus explicit methods work by advancing the kinematic state from a previous increment. By referring back to (3.17), the key components of this equation are the velocity and acceleration. For the implicit method, the mass and damping factors are ignored because the displacement is not a function of time. For the implicit method, the velocities and accelerations are zero, because the derivatives of the displacement are zero. Within this project certain models require very large global stiffness matrices to be inverted, and the computational time and the random access memory size requirement for this is significant. For implicit analysis methods such as the Newton-Raphson method are used to solve the finite element problem. The explicit method considers the

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solution to be a function of time, therefore both velocity and acceleration are included which means mass and damping effects are considered. For the explicit method a central difference method time integration scheme is used. The scheme calculates the field variables at the nodal points and the method works by inverting the lumped mass matrix. The basis of the solution works on time step increments. It is not possible to directly solve modal frequencies and mode shapes with the contact conditions modelled within this problem due to nonlinearities and varying values within the stiffness matrix. This can not be dealt with by the finite element method's eigenvalue solvers unless the contact conditions are ignored.