The Finite Element Method Computer Science Essay

Published Date: 02 Nov 2017

Chapter 2

Finite element method (FEM) is a numerical method for solving a differential or integral equation. It has been applied to a number of physical problems, where the governing differential equations are available. The method essentially consists of assuming the piecewise continuous function for the solution and obtaining the parameters of the functions in a manner that reduces the error in the solution

In the finite element method, the actual continuum or body of matter, such as a solid, liquid, or gas, is represented as an assemblage of subdivisions called finite elements. These elements are considered to be interconnected at specified joints called nodes or nodal points. The nodes usually lie on the element boundaries where adjacent elements are considered to be connected. Since the actual variation of the field variable (e.g., displacement, stress, temperature, pressure, or velocity) inside the continuum is not known, we assume that the variation of the field variable inside a finite element can be approximated by a simple function. These approximating functions (also called interpolation models) are defined in terms of the values of the field variables at the nodes. When field equations (like equilibrium equations) for the whole continuum are written, the new unknowns will be the nodal values of the field variable. By solving the field equations, which are generally in the form of matrix equations, the nodal values of the field variable will be known. Once these are known, the approximating functions define the field variable throughout the assemblage of elements. The solution of a general continuum problem by the finite element method always follows an orderly step-by-step process. With reference to structural problems, the step-by-step procedure can be stated as follows:

2.1.1 Discretisation of the structure

The first step in the finite element method is to divide the structure or solution region into subdivisions or elements. Hence, the structure is to be modelled with suitable finite elements. The number, type, size, and arrangement of the elements are to be decided.

2.1.2 Selection of a proper interpolation or displacement model

Since the displacement solution of a complex structure under any specified load conditions cannot be predicted exactly, we assume some suitable solution within an element to approximate the unknown solution. The assumed solution must be simple from a computational standpoint, but it should satisfy certain convergence requirements. In general, the solution or the interpolation model is taken in the form of a polynomial.

2.1.3 Derivation of element stiffness matrices and load vectors

From the assumed displacement model, the stiffness matrix [K (e)] and the load vector P (e) of element e are to be derived by using either equilibrium conditions or a suitable variation principle.

2.1.4 Assemblage of element equations

Since the structure is composed of several finite elements, the individual element stiffness matrices and load vectors are to be assembled in a suitable manner and the overall equilibrium equations have to be formulated as is the assembled stiffness matrix, is the vector of nodal displacements, and is the vector of nodal forces for the complete structure.

2.1.5 Solution for the unknown nodal displacements

The overall equilibrium equations have to be modified to account for the boundary conditions of the problem. After the incorporation of the boundary conditions, the equilibrium equations can be expressed as for linear problems, the vector can be solved very easily. However, for nonlinear problems, the solution has to be obtained in a sequence of steps, with each step involving the modification of the stiffness matrix [K] and/or the load vector.

2.1.6 Computation of element strains and stresses

From the known nodal displacements, if required, the element strains and stresses can be computed by using the necessary equations of solid or structural mechanics. The terminology used in the previous steps has to be modified if we want to extend the concept to other fields. For example, we have to use the term continuum or domain in place of structure, field variable in place of displacement, characteristic matrix in place of stiffness matrix, and element resultants in place of element strains.

2.2 Nonlinear Finite Element Analysis [14]

Non-linear finite element analysis is an essential component of computer aided design. Testing of prototypes is increasingly being replaced by simulation with non-linear finite element methods because this provides a more rapid and less expensive way to evaluate design concepts and design details. For example, in the field of automotive design, simulation of crashes is replacing full scale tests, for both the evaluation of early design concepts and details of the final design, such as accelerometer placement for airbag deployment, padding of the interior and selection of materials and component cross sections for meeting crashworthiness criteria. In many fields of manufacturing, simulation is speeding the design process by allowing simulation of processes such as sheet metal forming, extrusion and casting. In the electronics industries, simulation is replacing drop tests for the evaluation of product capability.

Nonlinear analysis consists of following steps

Development of model;

Formulation of governing equations;

Discretization of the equations;

Solution of the equations;

Interpretation of results;

Each of the steps in nonlinear analysis has an important role for a successful analysis of any structure. The following is a detail description of each step.

2.2.1 Development of model

Modelling is a term that tends to be used for two distinct tasks in engineering. The objective in this approach is to identify the simplest model which can replicate the behaviour of interest. In this approach, model development is the process of identifying the ingredients of the model which can provide the qualitative and quantitative predictions. A second approach to modelling, which is becoming more common in industry, is to develop a detailed, single model of a design and to use it to examine all of the engineering criteria which are of interest. The impetus for this approach to modelling is that it costs far more to make a model or mesh for an engineering product than can be saved through reduction of the model by specializing it for each application. By using the same model for all of these analyses, a significant amount of engineering time can be saved. The finite element model may serve as a prototype that can be used for checking many aspects of a design’s performance. The decreasing cost of computer time and the increasing speed of computers make this approach highly cost-effective.

2.2.2 Formulation of Governing equations

The formulation of the governing equations and their discretisation is largely in the hands of the software developers today. However, a user who does not understand the fundamentals of the software faces many difficulties, for some approaches and software may be unsuitable. To convert experimental data to input, the user must be aware of the stress and strain measures used in the program and by the experimentalist who provided material data. The solution of the discrete equations also presents a user with many choices. An inappropriate choice will result in very long run-times which can prevent him from obtaining the results within the time schedule. The approximate computer time required for various solution procedures are invaluable in the selection of a good strategy for developing a reasonable model and selecting the solution procedure. Governing matrix equation for nonlinear problem, only numerical solutions are possible. LS-DYNA uses the explicit central difference method to integrate the equation of motion. The semi-discrete equations of motion at time n is given as

Mn = Pn - F n + Hn (2.1)

Where, Mn is the diagonal mass matrix,

pn accounts for external and body force loads,

Fn is the stress divergence vector, and

Hn is the hourglass resistance.

2.2.3 Mesh description

In solid mechanics, Lagrangian meshes are most popular. Their attractiveness stems from the ease with which they handle complicated boundaries and their ability to follow material points, so that history-dependent materials can be treated accurately. In Lagrangian meshes, the nodes and elements move with the material. In the development of Lagrangian finite elements, two approaches are commonly taken:

Formulation in terms of the Lagrangian measures of stress and strain in which derivates and integrals are taken with respect to the Lagrangian (material) coordinates X, called total Lagrangian formulations.

Formulation expressed in terms of Eulerian measures of stress and strain in which derivates and integrals are taken with respect to the Eulerian (spatial) co-ordinates x, called updated Lagrangian formulations.

The accuracy and stability of solutions are important issues in nonlinear analysis. These issues manifest themselves in many ways. A judicious selection of an element considering the stability of the element, the expected smoothness of the solution and the magnitude of deformation expected. In addition, the analyst must be aware of the complexity of nonlinear solutions. The possibility of both physical and numerical instabilities must be kept in mind and checked in a solution.

It is important to know limitations of the different meshes in applications. Lagrangian meshes are generally used for the analysis problem where the deformation is not large. This type of problem is much common in impact dynamic analysis and crash test analysis. However, to widen the applications of the available Eularian and Lagrangian for finite element analysis, a combined Eularian-Lagrangian mesh is used. These are called ALE methods: arbitrary Lagrangian Eulerian. The aim of ALE finite element formulations is to capture the advantages of both Lagrangian and Eulerian finite elements while minimizing the disadvantages. This takes care of both large and small deformations, found in fluids and structures respectively.

Descritisation of Equations

The discrete equations for a finite element approximation will then be derived. For problems in which the accelerations are important (often called dynamic problems) or those involving rate-dependent materials, the resulting discrete finite element equations are ordinary differential equations (ODEs). The process of discretising in space is called a semi-discretisation since the finite element procedure only converts the spatial differential operators to discrete form; the derivates in time are not discretised .For static problems with rate independent materials, the discrete equations are independent of time, and the finite element discretisation results in a set of nonlinear algebraic equations.

For transient processes and solutions we use both implicit and explicit integration procedures; continuum procedures for equilibrium problems are considered. Newton methods and the linearization procedures required for the construction of the Newton equations are developed. In the solution of nonlinear problems, the stability of numerical procedures and of the physical processes is crucial. Therefore both geometric and material is considered.

For static problems with rate-dependent material behavior, the discrete equations are independent of time, and the finite element discretizations results in a set of nonlinear algebraic equations, which can be solved by the Newton-Raphson method.

Two-node linear order and three-node quadratic order element formulations are available.

C:\Users\Manjunath\Desktop\LagrangianMesh.png

Figure 2.1 Lagrangian meshes

Solution Methods

Implicit time integration

The unconditional stability of implicit integrators has not been proven for all nonlinear systems, although results which deal with specific situations indicate that unconditional stability holds at least for certain nonlinear systems. In any case, experience indicates that the time steps which can be used with implicit integrators are much larger than those for explicit integration in many problems. The major restrictions on the size of time steps in implicit methods arise from accuracy requirements and the decreasing robustness of the Newton procedure as the time step increases. The latter is particularly pronounced in problems with very rough response, such as contact-impact. With a large time step, the starting iterates may be far from the solution, so the possibility of failure of the Newton method to converge increases. The construction of the linearised algebraic equations for the Newton procedure is often quite involved. Furthermore, the storage of these equations requires significant amounts of memory. The memory requirements can be reduced substantially by iterative linear equation solvers (an iterative method within an iterative Newton method).

Figure 2.2 Graphical resprestation of explicit time integration

Explicit time integration

Explicit refers to the numerical method used to represent and solve the time derivatives in the momentum and energy equations. The following figure presents a graphical description of explicit time integration.

The displacement of node n2 at time level t+Δt is equal to known values of the displacement at nodes n1, n2, and n3 at time level t.

A system of explicit algebraic equations is written for all the nodes in the mesh at time level t+Δt. Each equation is solved in-turn for the unknown node point displacements.

Explicit methods are computational fast but are conditionally stable.

The time step, Δt, must be less than a critical value or computational errors will grow resulting in a bad solution.

Explicit analysis solves using Central difference method

The time step must be less than the length of time it takes a signal travelling at the speed of sound in the material to traverse the distance between the node points.

In the central difference scheme, the velocities are approximated by

(2.2)

Equation 2.3 represents the accelerations

(2.3)

i.e., the value of the derivative at the centre of a time step is obtained from the difference of the function values at the ends of the time interval, hence the name central difference formula.

2.3 Contact-Impact problem in general

Many problems in the simulation of prototype tests and manufacturing processes involve contact and impact. For example, in the simulation of a drop test, the components must be separated by so called sliding interfaces which can model contact, sliding and separation. In the simulation of manufacturing processes, sliding interfaces are also important, the modelling of the tool work piece interface in machining and the modelling of extrusion are some examples of where sliding interfaces are needed. In crash simulation of automobiles, many components, including the engine, wheels, radiator, etc., can contact during the crash and their surfaces are treated as sliding interfaces. The treatment of impact always requires a subsequent treatment of contact, since bodies which impact will stay in contact until rarefaction waves result in release. In the governing equations for bodies in contact kinetic and kinematic conditions on the contact interface are to be considered. The key condition is the condition of impenetrability: namely, the condition that two bodies cannot interpenetrate. The general condition of impenetrability cannot be expressed as a useful equation, so several approaches to developing specialized forms of these conditions have evolved.

Contact-impact problems are among the most difficult nonlinear problems because the response in contact-impact problems is not smooth. The velocities normal to the contact interface are discontinuous in time when impact occurs. When Coulomb friction models are used, the tangential velocities along the interface are discontinuous when stick-slip behavior is encountered. These characteristics of contact-impact problems introduce significant difficulties in the time integration of the governing equations and impair the performance of numerical algorithms. Therefore, the appropriate choice of methodologies and algorithms is crucial in the successful treatment of these problems. Techniques such as regularization are highly useful in obtaining robust solution procedures, but the analyst must understand their effect so that important aspects of the response are not eliminated.

LS-DYNA Capabilities

LS-DYNA is a general purpose FEA software that is ideally suited for the solution of complex non-linear dynamics and quasi-static problems, especially those involving Impact, contact and other highly discontinuous events. It supports stress-displacement analysis as well as fully coupled physics analyses, such as coupled temperature- displacement and coupled fluid-structural analyses.

The nonlinear dynamic analysis capability in LS-DYNA offers a unique combination of structural and material flow analysis, including structure to structure contact and coupled fluid-structure interaction analysis.

It uses proven explicit time integration technique to solve the nonlinear dynamic response analysis problem. It contains a complete library of finite elements (type/shapes/orders) and numerous material models to simulate the nonlinear behaviour of materials (metals/composites/plastic/foam). Geometric, material and contact non-linearities can be accommodated, as well as the complete failure of materials and structural parts. Contact between structural parts can be included; including the simulation of thin walled structure bending and flowing onto them.

For material flow modelling, it uses classic Eulerian frame of reference in which the finite element mesh does not distort; but remains fixed in space. The material simply flows through the Eulerian mesh from one element to next, while the elements themselves do not deform. This approach avoids the numerical difficulties often associated with the large deformation of FE meshes.

The Eulerian formulation in LS-DYNA allows the modelling of not only liquids and gases but also structural material like steel. This extends material flow analysis to more complex impact and penetration problems.

LS-DYNA Applications

Automotive Crashworthiness & Occupant Safety

LS-DYNA is widely used by the automotive industry to analyse vehicle designs. It accurately predicts a car's behaviour in a collision and the effects of the collision upon the car's occupants. Automotive companies and their suppliers can test car designs without having to tool or experimentally test a prototype, thus saving time and expense.

The non-linear dynamic analysis capability in LS-DYNA offers a unique combination of structural and material flow analysis, including structure to structure contact and coupled fluid interactions analysis.

One of LS-DYNA's most widely used applications is sheet metal forming. LS-DYNA accurately predicts the stresses and deformations experienced by the metal, and determine if the metal will fail. LS-DYNA supports adaptive remeshing and will refine the mesh during the analysis, as necessary, to increase accuracy and save time.

Metal forming applications include: Metal stamping , Hydro-forming, Forging  and Deep drawing. 

Aerospace applications include: Blade containment, Bird strike (windshield, and engine blade)  and Failure analysis.

LS-DYNA element library comprises of continuum, structural, inertial, rigid, capacitance, connectors, cohesive, spring, dashpot, and special purpose elements. Prescribed conditions in LS-DYNA include amplitude curves, initial conditions, boundary conditions and loads. Kinematic constraints, contact modelling re unique capabilities in LS-DYNA.

Specialized capabilities for modelling airbags, seat belts and sensors have tailored LS-DYNA for applications in the automotive industry. Adaptive meshing is available for shell elements and is widely used in sheet metal stamping simulations.

LS-DYNA's potential applications are numerous and can be tailored to many fields. It is a general-purpose multi-physics simulation software package and is not limited to any particular type of simulation. In a given simulation any of its many features can be combined to model a wide range of physical events. LS-DYNA is one of the most flexible finite element analysis software packages available.

Other applications include: Drop testing, Can and shipping container design, Electronic component design, Glass forming, Plastics, mold, and blow forming Biomedical, Metal cutting, Earthquake engineering, Failure analysis, Sports equipment (golf clubs, golf balls, baseball bats, helmets) and Civil engineering (offshore platforms, pavement design)

Contact-Impact problem in LS-Dyna

In LS-DYNA, a contact is defined by identifying (segments, parts, part sets etc.) location that to be checked for potential penetration of a slave node through a master segment. A search for penetration of a slave node through a master segment. A search for penetration, using any of a number of different algorithms, is made every time step. In the case of a penalty-based contact, when a penetration is found proportional to the penetration depth is applied to resist, and ultimately eliminate, the penetration. The contacts used are penalty-based contacts as opposed to constraint-based contacts.

Contact-Impact algorithms implemented in general purpose nonlinear finite element analysis software can treat the interaction of many bodies, but multi-body contact consists of the interaction of pairs of bodies. The appropriate choice of methodologies and algorithms is crucial to success, and regularization techniques are highly useful in obtaining robust solution procedure.

2.5 Finite Elements

2.5.1 Belytschko-Lin-Tsay shell element

The Belytschko-Lin-Tsay shell element was implemented in LS-DYNA as a computationally efficient alternative to the Hughes-Liu shell element. For a shell element with five through thickness integration points, the Belytschko-Lin-Tsay shell elements requires 725 mathematical operations compared to 4050 operations for the under integrated Hughes-Liu element. The selectively reduced integration formulation of the explicit Hughes-Liu element requires 35,350 mathematical operations. Because of its computational efficiency, the Belytschko-Lin-Tsay shell element is usually the shell element formulation of choice. For this reason, it has become the default shell element formulation for explicit calculations.

The Belytschko-Lin-Tsay shell element figure (3.3) is based on a combined co-rotational and velocity-strain formulation. The efficiency of the element is obtained from the mathematical simplifications that result from these two kinematical assumptions. The co-rotational portion of the formulation avoids the complexities of nonlinear mechanics by embedding a coordinate system in the element. The choice of velocity-strain or rate-of-deformation in the formulation facilitates the constitutive evaluation, since the conjugate stress is the physical Cauchy stress.

2.5.2 Solid Element (8-noded brick element)

It is used for 3-D modelling of solid structures. The element is defined by eight nodes having 3 Degree of Freedom at each node: translation in the nodal X, Y, Z directions.

The element has plasticity, creep, swelling, stress stiffening, large deflection and large strain capabilities. A reduced integration option with hourglass control is available. The geometry, node locations and co-ordinate system for this element are shown in figure 3.4. The element is defined by 8 nodes and orthotropic material properties. Orthotropic material directions correspond to the element co-ordinate directions. Pressure may be input as surface loads on the element faces as shown by circled numbers. Using uniform reduced integration provides the following advantages when running non-linear analysis.

Less CPU time is required for element stiffness formation and stress/ strain calculation to achieve a complete accuracy to the full integration option.

Analysis will not suffer from volumetric locking which can be caused by plasticity or other incompressible material properties.

C:\Users\Manjunath\Desktop\REPORT\SNAPS\TB.png

Figure 2.3 Typical 4-noded Belytschko-Tsay shell element

Figure 2.4 8-noded brick element

2.6 Time step calculations

2.6.1 Time step calculations for shell elements

For the shell elements, the time step size is given by

(3.9)

Where, LS is the characteristic length and C is the sound speed

(3.10)

Three user options exist for choosing the characteristic length. In the default or first option the characteristic length is given by

(3.11)

Where β0 for quadrilateral and 1 for triangular shell elements, AS is the area, and Li, (i= 1 . . . . 4) is the length of the sides defining the shell elements. In the second option a more conservative value of L s is used

(3.12)

Where Di ( i=1, 2) is the length of the diagonals. The third option provides the largest time step size and is frequently used when triangular shell elements have very short altitudes.

LS is given by,

(3.13)

2.6.2 Time Step Calculations for Solid Elements

A critical time step size, ∆te, is computed for solid elements from

(3.14)

Where Q is a function of the bulk viscosity coefficients C0 and C1

(3.15)

Le is a characteristic length:

(3.16)8-node solids:

4-node tetrahedras:

Ê‹e is the element volume, Aemax is the area of the largest side, and c is the adiabatic sound speed.

(3.17)

Where ρ is the specific mass density. Noting that

(3.18)

And that along an isentrope the incremental energy, E, in the units of pressure is the product of pressure, p, and the incremental relative volume, (dV)

(3.19)

We obtain

(3.20)

For elastic materials with a constant bulk modulus the sound speed is given by:

(3.21)