Jaws : Finite Element Method

Title: Finite Element Method

Central Concept: The finite element methond - dividing a complex problem into many simple problems

Learning Standards (National Science Education Content Standards):

B.4 - Motions and forces

A.1.4 - Formulate and revise scientific explanations and models using logic and evidence

A.1.1 - Identify questions and concepts that guide scientific investigations

Essential Questions:

How do materials respond to forces?

What are the tension, compression, and shear loads?

What is stress?

How can we estimate deflection of complex shapes?

Content and Skills:

Develop a understanding of the relationship between force and deflection (B.4)

Understand methods for dividing complex problems into many simple problems (B.4)

Scientific inquiry and reading comprehension (A.1.4, A.1.1)

Note: The goal of this lesson is to introduce the students to the concepts of deflection (strain), stresses, and the finite element method. This is meant to motivate students by giving them a glimpse of subjects which are outside of the typical high school curriculum.

Learning Activities:

• Looking at the image comparing a gharial with an alligator, which one looks to have the stronger jaw, why?
  • Look at the gharial and the alligator in more detail.
    • Their bodies are of similar sizes, but their skulls and jaws are very different. Their jaws are as different as their diets: the gharial eats fish while the alligator eats a large variety of prey including animals as large as deer.
• Although the alligator jaw looks stronger than the gharial's jaw we need a more conclusive understanding.
• To understand how structures respond to forces, we use the theory of continuum mechanics.
  • Continuum mechanics tells us that materials deform when we push or pull on them.
  • Forces and material deformations are grouped into 3 types (picture):
    • Compression (a)
    • Tension (b)
    • Shear (c)
  • When materials experience a load they deform, and the amount of deformation is a property of the material, it is called stiffness. Materials with a greater level of stiffness deform less when they experience a force.
  • All things being equal, larger structures are stronger than smaller ones. For example, when comparing two similar ropes made from the same materials, the thicker rope can hold more weight. In order to take into account the differences in size we have the concept of stress, which is the force applied divided by the area through which it "flows". Continuing with the rope example, we can expect that if the cross sectional area of a rope were to double, the force would also have to double in order to maintain the stress level.
  • In reality, shapes are rarely as simple as the shapes in this picture. Over time physicists have developed equations for estimating the stresses some commonly used structural shapes. These are shapes that can be readily described using mathematical expressions, such as circles, rectangles, and ellipses. Examples of these shapes can be seen on the Wikipedia page on moments of inertia
• What about complex irregular shapes like the jaw of an alligator?
  • Can we draw a lesson from out study of the motion of thelizard jump?

Using computers to estimate the strength of crocodilian jaws

• Our goal is to compare the strength of the gharial jaw to the alligator jaw.
• Like in the jumping lizard example we have a complex problem. Like before we are going to divide the problem into many simple problems which can be solved with the aid of computer.
• To have a good comparison of the jaws, we are going to compare the amount of deformation the skulls suffer when loaded with the same force.
  • We are going to subject both skull to the same biting muscle force and then compare the amount of the deflection the skulls experience.
  • Our goal is to compare, not to measure actual deflections.
• Calculating the deflection of simple shapes can be easily calculated, but the deflection of a shape as complex  as an alligator jaw is very difficult to calculate.
  • Like the lizard jump example where the flight of the lizard was divided into many small intervals where the motion was simplified. The crocodilian jaw can be approximated as many simple shapes stuck together in a mesh (Figures 3 and 4). The deflection of each section can then calculated and combined to get an estimate of the deflection of the complex structure.
• The formal method for analyzing complex structures, by dividing them into small and simple element is called the Finite Element Method. The analysis of structures using the finite element method is called finite element analysisAnalysis of the physical behavior of a finite element model. In this analysis a given physical "treatment" is applied (such as force loading on some elements), followed by computation of the effects of this treatment on other elements of the FEM. More specifically, the analysis phase involves solving a set of simultaneous algebraic equations in which the unknown variables that are solved for in this system of equations are the unknown nodal degrees of freedom. Once the values of the unknown nodal degrees of freedom are found, the unknown reaction forces are determined. Next, the spatial variation of the primary field variables within each element is computed using the element's predefined interpolating polynomials and the element’s nodal values. Thus, for solid elements this results in mathematical functions that completely define the displacement field within every finite element. These functions are then differentiated to obtain the complete strain field within each element which, when combined with known material properties of the element, yields the element stress field. In summary, the finite element solution will yield 1) the reaction forces necessary to maintain static equilibrium of the system, 2) the displacement field ( i.e. displacements of the material through out the 3-D domain), the complete strain tensor field, and the complete stress tensor field. (FEA).
• The finite elementsa subdivision within the material that is relatively simple in shape, such as a tetrahedral or hexahedral, defined by vertices called nodes. method is widely used in engineering to predict the behavior of complex structures.
• The steps needed to conduct FEA are the following:
  1. Create finite element modelA computer-based model of a physical object or system composed of contiguous set of finite elements and associated nodes that model the volume of the object. The finite elements are assigned material properties such as elasticity, thermal conductivity, etc. to account for the material behavior associated with the physical phenomenon being modeled. A finite element model (FEM) is the dataset used for finite element analysis (FEA). Individual finite elements may have different material properties, loading conditions, size, shape, etc., enabling the most complicated physical systems to be modeled.
     • Model - Create a shape that represents the structure being analyzed.
     • Mesh - Divide the complex shape into simple elements.
     • Load - Define forces which simulate the forces which the structure might experience in use.
     • Example images: Meshed alligator, Meshed gharial
  2. Solve
     • Using a solver, solve the finite element problem in order to get stresses and deflection for the elements in the model
  3. Process the results
     • Combine the results such that the stresses and deflections can be visualized and understood.
     • Example images: Deformed meshed alligator, Deformed meshed gharial 
     • Example animated deformations: Alligator, Gharial
       

Understanding the results

• The images, Deformed meshed alligator, Deformed meshed gharial, show an exaggerated representation of the deflection which a the skulls of alligators and gharials might experience if they are subjected to the same biting force.
• What does this tell you about the strength of their jaws?
  • The gharial jaw deflects or bends much more than the alligator jaw. Is this okay? Is the gharial at a disadvantage?
  • Not necessarily, the gharial's jaw might not be exposed to the same forces as an alligator.
    • Gharials eat almost exclusively fish, which do not require the same amount of force to capture, kill, and eat as would be required by an alligator as it tries to catch an antelope.

Equipment / Materials:

Internet access – projector or computer lab

Expected Outcomes:

Students should:

1. Have basic understanding of force-deflection relationship
2. Have basic understanding of stress
3. Have basic understanding of what is the finite element method

Assessment / Assignment(s):

Homework:

Interactive assessment - classroom discussion.