As we know, finite element analysis is a very powerful tool to numerically analyze the behavior of any physical or mechanical system. In this method, the entire computational domain is divided into smaller elements, within which the equations of motion are approximated, and boundary conditions are applied to solve for the unknown field variables like displacement, velocity, strain, and stress.Â
Meshing (Discretization) Basics
Each element in FEA is described by a set of basis functions also called shape functions. These functions specify how the field variables such as displacement change within each element. The real-life engineering problems come in different shapes and sizes. So, during the discretization process of the domain, there will be lot variability in shape and size of the elements, and it is far from ideal to derive shape functions and numerically integrate for each element separately. This process is simplified by using isoparametric mapping, which involves having one natural element for each geometric shape (1D, 2D and 3D).
A local natural coordinate system is attached to this element only based on its geometry and, subsequently, shape functions are derived and integrated. There is a relationship, known as transformation mapping, between natural elements and actual physical elements. Using this relationship, the natural formulations are then transformed based on the physical element size in the FEA problem. All of these precalculated formulations for various element libraries are stored in commercial FEA software like Abaqus. So, when we pick a certain element type in our model to mesh the geometry, we are mapping our physical element back to a natural element stored in the library. In this article, let’s first understand different types of natural elements, followed by the basics of mapping process using a simple 1D axial element.
Different Types Of Natural Elements
Now let’s take a look at a few types of natural elements.
Linear 1D Natural element
Quadratic 1D Natural element
Linear 2D Natural element
Quadratic 2D Natural element
Linear 3D Natural element
Process Of Isoparametric Mapping
Now, let’s briefly discuss the mapping process using a simple linear 1D axial bar element.
The mapping primarily uses the two relationships shown below,
Where {u}, {x} are displacements and coordinates at any point within the element respectively. {d} and {X} are the nodal displacements and coordinates of the element. [N(s)] is the shape function matrix of the element represented as a function of natural coordinate system.
For the 1D bar, the shape function matrix is derived to be,
The mapping function called the Jacobian (J(s)) is calculated for the 1D bar problem as shown below,
In 1D element, the Jacobian physically represents the shrink or strength ratio of the natural elements at a point ‘s’ of the physical domain.
Uniform And Non-Uniform Mapping
In uniform mapping, the natural element is simply expanded or compressed to obtain the physical element. In this scenario, we get a constant Jacobian.
In non-uniform mapping the shape and size of the element changes considerably. Here, the Jacobian will be a function of s.
We often see element distortion errors while running our FEA models. This error is coming during the mapping process of physical element to natural element. When J(s) <0, then the element segment at ‘s’ is flipped over on the physical element causing the error. To avoid this, we must use good quality elements while meshing the computational domain.
Advantages Of Isoparametric Mapping
Irregular domains, curved domains and intricate geometrical features can be modeled easily using the same set of shape functions as the natural element to calculate field variables. This adaptability is indispensable in real-world engineering simulations, which often exhibit complex geometry.
The variation of material properties of an element can be properly accounted for by using appropriate values at the element gauss points.
This also facilitates the use of higher-order elements, allowing for more accurate representation of the solution within each element.
Since the isoparametric mapping allows for the use of same set if shape functions for different sizes and shapes (within certain bounds), this allows for an efficient process of numerical integration over each element.
Final Thoughts
Hopefully, this article has provided a brief understanding of the concept of isoparametric mapping and its application in commercial FEA. This is a fundamental principle in FEA that ensures consistency, accuracy and efficiency in the analysis process and its adoption has greatly enhanced the capability of FEA software.
If you're interested in learning more about Abaqus or FEA in general, don't hesitate to get in touch with our team!