# What is Nonlinear FEA and What Attributes to the Source of Nonlinearity?

February 3, 2020

If the definition of the word nonlinear is carefully examined, you might mistake it for the description of life. Take, for example, one dictionary’s definition of nonlinear: not denoting, involving, or arranged in a straight line. However, it is the second definition of nonlinear that I believe captures the closest parallel to a description of life: not linear, sequential, or straightforward; random.

Furthermore, it is one word in particular that draws this correlation, and more or less describes life; that word is random. Even conventional wisdom agrees that evolution occurs (in part) through the course of random mutations. Moreover, these evolutionary changes allow an individual organism to be better able to survive as well as adapt per natural selection.

When you’re attempting to analyze nonlinear processes, it might seem difficult-to-impossible at first; however, there are simulation tools which allow accurate modeling of nonlinear behaviors, especially within electronics. Nonlinear FEA or FEM simulation tools can be of great service to you and your high-risk designs.

## What is Finite Element Analysis or FEA?

Per its definition, FEA (Finite Element Analysis) is a simulation of any specified physical phenomenon using a numerical technique called FEM (Finite Element Method). Furthermore, Engineers currently use it to limit the number of experiments and physical prototypes while also maintaining component optimization. This, of course, aids in the design phase of their products while also improving product development speed, general product development, and overall product quality.

Moreover, mathematics is essential to the comprehensive understanding and quantification of any specified physical phenomena. These physical phenomenons include:

• Thermal transport

• Fluid behavior

• Structural behavior

• Wave propagation

• Biological cell growths

Furthermore, the processes listed above utilize PDEs (Partial Differential Equations) to provide an accurate description of each specific process as a whole. Although we use a computer for solving these PDEs, the most renowned numerical technique in use today is the FEA.

## The Role of Differential Equations in FEA

The role of differential equations includes describing the processes of nature as well as physical phenomena within the field of engineering mechanics. Also, solving these PDEs (complex equations) provides the ability to compute relevant structural quantities of a structure such as strains and stresses. This, in turn, affords an accurate estimation of the behavior of the component under test while under a specified load.

Also, there are two facts one needs to keep in mind in regards to FEA. Firstly, Finite Element Analysis only provides an approximate solution to a given problem. Secondly, FEA is the numerical approach to obtaining real results to these PDEs. In summary, Finite Element Analysis is a mathematical method in use to predict how an assembly or part will behave under specified conditions.

Furthermore, the use of FEA provides the basis or foundation for modern simulation software as well as assists engineers in locating areas of tension and weak spots within their designs. In general, the results of these FEA-based simulations are physically represented by a color scale that displays the pressure (distribution) over a particular object. In essence, FEA provides the ability to estimate the aggregate effects of small changes.

## The Two Types of Finite Element Analysis or FEA

In summary, there are two types or categories of FEA, one is called Linear, and the other is called Nonlinear FEA. In terms of a definition, Linear means extending along a straight or nearly straight line or arranged in a straight or almost straight line. Furthermore, Linear can also mean linear movement or progressing from one stage to another in a single series of steps, sequential.

FEA analysis attempts to segment shapes into smaller and smaller polygons until the entirety of the object can be mapped.

Moreover, in terms of Linear FEA, it refers to simple Linear behavior such as that of a linear function. In other words, Linear FEA refers to an analysis where the linear relation between displacements and applied forces remains true. In a real-world scenario, this applies to structural issues where the stresses persist in the linear elastic range of the material in use.

However, as established earlier, Nonlinear is, in essence, random. In regards to Nonlinear FEA, it refers to the nonlinear behavior of a material, contacts in assemblies, and geometry.

## Nonlinear FEA in More Detail

As I am sure you are aware, the majority of issues either start out being nonlinear, or they evolve into nonlinear once they reach increased load levels. Furthermore, if an issue is typified by large deflections in comparison to the problem’s dimensions, then it involves geometric nonlinearity. Moreover, if the relationship between the strain and the stress becomes nonlinear, for example, plasticity, then the issue involves material nonlinearity.

In summary, various system properties attribute to the source of nonlinearity, such as geometry, materials, constraints, and nonlinear loading. Here are some of the examples in more detail.

### Geometric Nonlinearity

Whenever an analysis involves geometric nonlinearity, the changes in geometry as the structure deforms is a consideration in the formulating of the equilibrium and constitutive equations. Furthermore, multiple engineering applications, for example, medical device analysis, and metal forming, necessitate the use of significant deformation analysis founded on geometric nonlinearity. Also, small deformation analysis established on geometric nonlinearity is a requirement for specific applications, such as those involving cables.

### Material Nonlinearity

In regards to material nonlinearity, it involves the nonlinear behavior of materials based on pressure, deformation history, current deformation, temperature, or rate of deformation. Examples of nonlinear material models include large strain hyperelasticity and elastoplasticity.

### Constraint and Contact Nonlinearity

When kinematic constraints are present in the model, constraint nonlinearity in a system occurs. Furthermore, it is possible to constrain the kinematic degree-of-freedom of a model through the use of restrictions on its movement.

Note: Kinematics is a branch of traditional mechanics that describes the motion of objects, bodies, points, as well as groups of objects or systems of bodies, without consideration of the forces promoting them to move.

Segmented analysis helps allow for more accurate simulation and modeling results.

With linear FEA, it is a rather straightforward process since there exists a linear relation constant between applied forces and displacements. However, this particular approach will not provide the analytical accuracy for higher load levels. Therefore, it might be acceptable to use Linear FEA for an initial analysis, but in most cases, it will require a Nonlinear FEA to solve the issue accurately. In essence, Nonlinear FEA is a requirement for more complex issues since it provides the ability to estimate the aggregate effects of small changes.

Working through nonlinear designs can be difficult for any designer, as challenges can occur unresolvable in a predictable manner. Cadence’s suite of design and analysis tools are here to work with you through the design process to ensure you have the greatest security in your designs. And Allegro PCB Designer is the layout system that can take your designs and collaborate with any analysis or simulation tool to get your job done.

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