What Is Nonlinear Analysis? Overview, Types, and Differences
In engineering practice, problems often cannot be solved with a simple linear calculation. In structural mechanics, stresses are sometimes high, and plasticity (or other nonlinear material properties) must be considered. Sometimes, elements are slender and sensitive to changes in geometry; in such a case, geometric nonlinearity could be more appropriate. In this post, we will briefly describe what structural mechanics analysis means when using nonlinearity and show elementary examples of nonlinear problems.
Keep reading to learn about nonlinear structural analysis, how it differs from linear and linear static analysis, why nonlinear analysis is essential, and what type of approximation errors are induced by static and linear elastic analysis.
Before comparing linear and nonlinear behaviors and simulation methods, the article will introduce basic mathematical and physical concepts of stiffness matrices, material properties, and loading conditions. What factors determine whether it should be calculated with the linear or nonlinear assumption?
We will conclude by considering the use of AI in mechanical engineering.
What Is the Finite Element Method for a Structure?
Let us first introduce an essential software toolbox for engineers and advanced architects. The finite element method (FEM) or finite element analysis (FEA) is a numerical technique for solving a range of physical phenomena. Finite elements are a standard tool for structural problems that aren’t analytically treatable and thus require numerical approximations.
FEA tackles structural problems and thermal (or thermo-mechanical), electrical, magnetic, acoustic, or combinations of these (multi-physics). Software vendors provide FEA-dedicated software packages or bundles with major CAD products. However, in addition to this “classic” simulation workflow, the 3D deep learning application also impacts this application, as shown at the end of the article.
We will also focus on when a structure exhibits geometric or material nonlinearity.
What Is the Stiffness Matrix?
The primary characteristics of a finite element are embodied in the element stiffness matrix. For a structural finite element, the stiffness matrix contains the geometric and material behavior information indicating the element’s resistance to deformation when loading. Such deformation may include axial, bending, shear, and torsional effects. The term stiffness matrix is also used for finite elements in nonstructural analyses, such as fluid flow and heat transfer. When subjected to external influences, the matrix represents the element’s resistance to change.
But how does this matrix arise? Let us analyze and review the concept first derived from scalar equations:
If an object can be considered a connection of springs, in mathematical form, the system response “x” for any applied input “y” could be given using a scaling factor k as
k * x = y
This assumption is similar to the formulation of the spring equation where “k” is the spring stiffness, “x” is the spring displacement, and “y” is the applied force.
For any generic system, it can be written in matrix form as:
[𝐾] {𝑥} = {𝑦}
where:
- {𝑥} can be displacements, temperatures, etc.,
- {𝑦} is a force, flux, etc.
- [𝐾] is the scaling factor commonly known as a stiffness matrix.
The Linear and Non-Linear Analysis Approaches
Having assembled the basic concepts of the finite element method and linear analysis, we now define nonlinearity concisely yet comprehensively, focusing on boundary nonlinearity and structural nonlinear analysis applications later in the article.
Linear and Nonlinear Equations
Given two variables, x and y, the general representation of a linear equation is as a line in the XY space (hence the name of linear!)
y = a * x
where “a” is the slope of the line, and we assume the line intersects the XY origin where x=0 and y=0.
Always reasoning in purely elementary concepts of graphs in the XY plane, the following equation is non-linear:
y = a * sin(x)
In this case, depending on how near we are to the XY origin, the linear approach can be more or less valid as an approximation.
Basic Linearities in Structure Analysis
Linear FEM methods may not accurately capture behavior in highly nonlinear scenarios, such as large deformations, material yielding, or contact problems, however it is essential to understand linear problems before venturing into nonlinear analyses.
In geometric linear analysis, changes in geometry with loading conditions are infinitesimally small since the model experiences small deflections and deformations based on the applied forces.
Material linear analysis - here, a linear (elastic) material is characterized by a linear stress-strain relation
σ = E ε
In this linear regime, the stress-strain response is linear and defined by the expression below, which should be familiar (Hooke’s Law). E is defined as Young’s Modulus, a material constant that describes the material’s stiffness. Metals are typically in the 100s of GPa, ceramics are high in the 200s and 300s of GPa, and polymers are closer to 1 GPa.
Linear Dynamic Analysis - Modes of Structural Vibrations
We will briefly discuss how to estimate and predict the natural frequencies of your linear model. As an engineer or architect, you will find this helpful approach to prevent your structure from entering resonance, which may hurt!
How Nonlinear Structural Analysis Arises
Geometric nonlinearity occurs when the geometry of a structure or a component experiences large deformations, which can cause it to behave nonlinearly. A typical practical example of such a structure is a fishing rod.
Material nonlinearity occurs when the component exceeds the yield limit, and the stress/strain relationship becomes nonlinear as the material deforms permanently. Thus, the equilibrium stress/strain relationship becomes some nonlinear function:
σ = E ε --> σ = f(ε)
(Physics courses.lumenlearning.com)
Contact includes the effect when two components come into contact where they can experience an abrupt change in stiffness, resulting in localized material deformation at the contact region.
Essential Characteristics of Nonlinear Structural Analysis
We will now review the fundamental characteristics of non-linear analysis, such as sensitivity to initial conditions, feedback loops, and emergent behaviors.
We will also discuss the importance of non-linear analysis in understanding complex systems and phenomena and explore how it impacts fields like physics, biology, or economics.
Linear vs Nonlinear Analysis
Let us briefly outline the differences between linear and non-linear systems with a “fun” example and serious yet simple nonlinear mathematics.
For example, the Lotka-Volterra equation describes the dynamics of a system where two species interact (predator and prey); the model captures the oscillations in population sizes over time, for instance, tuna (density x with rate of change x’) and sharks (density of population y with rate of change y’):
x' = a * x - b * x * y
y' = c * x * y - d * y
In summary, linear equilibrium focuses on stability around fixed points using linearization, while nonlinear equilibrium considers the system’s behavior without simplifying it to a linear approximation.
Types of Nonlinear Structural Analysis
We return from tuna and sharks to engineering to briefly summarize the different nonlinear analysis types useful for structural engineers. Engineers must consider linear and nonlinear effects to ensure reliability.
The main advantage of nonlinear analysis is its ability to predict behavior accurately. It considers the effects of large displacements, material nonlinearity, and other interactions. By accounting for nonlinear factors, engineers can predict how a system responds under various loading conditions more realistically. For instance, in design for reliability, non-linear structural analysis provides realistic results for structures subjected to extreme forces or boundary conditions.
Nonlinear Analysis of Structures: The Arc Length Method
The arc length method extends the concepts of Newton’s method. Traditional methods like Newton’s method are not ideal for such cases because they may fail when dealing with structural instabilities like softening, buckling, and material failure.
Instead of incrementally solving concerning degrees of freedom (as in Newton’s method), the arc length method focuses on solving nonlinearity while considering the arc length along the equilibrium path. It ensures that the solution follows the nonlinear equilibrium path through limit points.
Challenges and Innovations for Nonlinear Analysis
Let us summarize the advantages and challenges of nonlinear analysis and propose a way to combine high-fidelity nonlinear analysis data with artificial intelligence techniques.
Computational Demands
Nonlinear analysis provides valuable insights into complex systems but demands computational resources and careful data handling. Researchers and design engineers must balance accuracy, efficiency, and interpretability when dealing with nonlinear phenomena, such as in the illustrated case of battery crash simulation.
Solving Nonlinear Analysis with Deep Learning
The intersection of nonlinear analysis (such as finite element analysis) and deep learning is an exciting area of research with valuable results for design engineers. At the same time, we maintain the assumption that skilled analysts will remain in charge of producing high-fidelity nonlinear analyses directly from FEA!
Deep learning is a relatively new branch of machine learning. In our case, it uses neural networks with multiple layers (deep architectures) and specialist filters and operations for efficient image recognition (max pooling, convolutions, etc.) to learn complex visual patterns from data and give them meaning either as labels (classification) or as continuous numerical values (regression).
This is why neural networks excel at tasks like image recognition, natural language processing, and even playing games (e.g., AlphaGo).
Deep learning models can automatically learn features, using them as robust pattern recognition and prediction tools. Data scientists and application engineers at Neural Concept explored using deep learning techniques to enhance nonlinear analysis:
1) Deep learning models can directly predict material properties (e.g., stress-strain curves) from raw material data, reducing the need for handcrafted constitutive models.
2) Training neural networks to provide a surrogate for the nonlinear analysis process (e.g., predicting stress distribution) with a speedup of 1.000X or even 100.000X compared to traditional simulation.
Appendix - Miscellaneous Information (Answers to FAQs)
Q. What is the linear superposition principle? Is it enough?
The linear superposition principle states that for all linear systems, the net response caused by two or more inputs is the sum of the responses that would have been caused by each input individually.
However, we must go beyond the linear superposition principle in dealing with nonlinear behavior.
Linear equations, such as those derived from Hooke’s law, break down when applied to highly deformed structures.
Instead, we use nonlinear equilibrium equations to account for material nonlinearity, geometric nonlinearity, and other factors. These equations relate internal forces, displacements, and external loads nonlinearly.
Q. What is the displacement control method in nonlinear analysis?
Rather than applying external loads incrementally, we prescribe the desired displacements at specific points in the structure. The equilibrium equations are then solved iteratively to find the corresponding internal forces.
This method is beneficial for analyzing structures with large deformations or undergoing plastic behavior.
Q. What simplified elements can we use in structural analysis engineering to model complex structures?
Beam elements are commonly employed to represent beams, columns, and other members. These elements account for axial, bending, and shear deformations. Additionally, we consider residual stresses, which exist within a material even when no external loads are applied.
These residual stresses can significantly affect the structural response.
Q. What is the importance of lateral displacements? How do we prevent them in practice?
Lateral displacements play a crucial role in assessing structure stability. When subjected to lateral loads (such as wind on bridges or cable structures or seismic forces on buildings), structures may experience buckling or sway. Analyzing them helps us determine whether a structure remains stable. Excessive lateral movement can lead to structural failure, especially in tall or slender structures.
Once analyzed, practical solutions such as lateral bracing systems (e.g., shear walls, cross-bracing, or moment frames) help control lateral displacements and enhance stability.
