We’ll continue on now with our blog series on finite element analysis (FEA).  After discussing how to best set up a computer-aided design (CAD) model for FEA simulation, in this blog I’ll cover the next step: meshing the model and applying boundary conditions.  “Meshing” is the process by which the CAD model is separated into discrete finite elements; it can be done in the same program that runs the FEA numerical simulation later, or it might be performed in a standalone program, depending on your software.  Boundary conditions are the loads (forces, movements, impacts, etc) and constraints that interact to actually cause deformation and stress in each element, and in turn the model as a whole.

# Mesh Generation

The mesh essentially gives finite element analysis its name; breaking a large complex shape into many smaller simple shapes allows the FEA program to easily evaluate the stresses for those simple shapes.  In a 3D element like our concrete slab in Figure 1, these elements might be simple cubes, or they might be more irregular pyramids and tetrahedrons.  Once the meshing program has evaluated the behavior of each discrete shape, it can integrate the data from these elements to create a model for the complex shape as a whole.

## Automatic versus Hand Generation

FEA can be performed by hand, as it was during its initial development in the 50s and 60s (Comini, p. 1)

[i].  Most FEA software still allows the user to develop meshes by hand, but for anything beyond the simplest shapes it becomes quite time consuming.  It must also be redone for each design iteration.

In general, it is preferable to allow a meshing algorithm to generate the mesh.  Based on a few parameters that the user can adjust, including ideal element size and aspect ratio, the program will move through the model and attempt to create a cohesive mesh.  However, despite many improvements over the years, automatic mesh generation is still not foolproof.  Even a simple flat plate can be have trouble meshing, if there are discontinuities in the boundary conditions.

## Guided Automatic Generation

Meshing programs usually offer tools to locally refine a mesh at specific spots on the model.  Autodesk Simulation, for instance, allows the user to create nodes on the model and then force the mesh to generate elements of a smaller size in a certain radius around that node.

The best solution for mesh generation is a compromise between drawing the mesh by hand and giving the meshing program total control.  An experienced FEA consultant can design or reconfigure a CAD model such that it encourages the mesh to form a certain way.  With an understanding of mesh generation algorithms, how meshes should be formed for certain constructions (for instance, at sharp corners or through thin plates), as well as how the algorithm interacts with the CAD file, an expert FEA engineer can make the meshing program run in predictable and useful ways.  Figure 1, at the opening of the blog, shows a portion of our final mesh after we’d guided its generation with some careful CAD work.  This is one reason that Glew Engineering keeps multiply CAD licenses, to correct CAD designs for FEA purposes.

# Boundary Conditions

FEA is the solution of partial differential equations for many small elements, with certain boundary conditions applied to the perimeter and perhaps internal nodes.  The boundary conditions are thus the loads.  FEA modeling is always concerned with how an object will respond to some external stimulus, simply called the loads.  Force on a part results in deformation, or motion.  If the part is static, then there must be reaction forces opposing the loads.  The concrete slab in this problem is being pulled down by gravity and the loads placed on the floor, but opposed be the support columns and constraints around the perimeter of the floor.  Without specifying both the applied loads and external constraints, the floor would not be static.

There are a few types of loads that a CAD model can be subjected to in a FEA program.  A force pushes or pulls on a specific section of the model, while a pressure exerts a distributed force across a surface.  An impact is simply a force that is exerted instantaneously but then drops to zero.  Objects can also be set with an initial velocity, to study collisions.  Enabling gravity pulls all of the elements downward equally.  Lastly, for thermal or electrostatic analyses, surfaces can be set to a specific temperature or exposed to an electrical current.

Gravity is essential in this simulation, since the main contributor to punching shear is the weight of the concrete slab itself.  In civil engineering, the weight of the structure is called the “dead load”.  There was also a “live load”, representing the people, furniture and equipment on top of the slab.  We modeled these loads with a pressure, or equally-distributed force, across the top surface of the slab.  The live load pressure is represented in Figure 2 as the orange force arrows pressing down on every element along the top surface.

## Constraints

Constraints prevent some part of the concrete slab from moving vertically, which causes the weight to deform the slab.  Every model needs a set of physical constraints keep it from flying off into infinity while still letting it bend, expand or contract as it would in real life.  These constrains can prevent translation (movement) in the x, y and z directions or rotation about the x, y and z axes.  Many times the constraints on a given surface will involve restraining some translational axes and some rotational axes.  If the model has been cut across a symmetry plane, for instance, then the elements on that surface need to be constrained such that they can’t move across it or rotate into it (otherwise our side and the mirrored side would rotate into each other, which we can’t allow).

The small red circles shown in Figure 2 mark the constraints on every node (the intersections between element) along one symmetry plane on our slab.  To obtain the correct symmetry behavior, we set:

• No translation in the x-direction
• No rotation about the y-axis
• No rotation about the z-axis

# Applying loads and constraints in FEA

The initial steps in setting up an FEA model determine the simulation.  The CAD model may need to be constructed in a certain way to allow the mesh to generate.  Then, that mesh needs to meet certain criteria if it’s going to be useful and give accurate results.  The loads need to be set correctly. Appropriate constraints ensure that the system is not under- or over-constrained.  Accurate and useful results in FEA simulation requires one to be careful and thorough in the initial setup.

Glew Engineering has done FEA consulting on a variety of projects, from semiconductor chambers and optical systems to vehicle lifts, heat exchanges and smart phone headsets.  Most of these we can’t post for proprietary reasons.  Take a look at our finite element analysis consulting services, and let us know how we might help you.

# References

[i] Comini, G., & Giudice, S. (1994). Finite element analysis in heat transfer: Basic formulation and linear problems. Washington, D.C.: Taylor & Francis.