Multiscale Modeling is a broadly used term to describe any instance where a physical problem is solved by capturing a system’s behavior and important features at multiple scales, particularly multiple spatial and(or) temporal scales. For instance, the picture below is a temporal multiscale representation of the origins of life.
Applications for multiscale analysis include: fluid flow analysis, weather prediction, operations research, and structural analysis - to name a few.
At MultiMechanics, our interest and expertise lies in the multiscale structural analysis of "advanced materials" - which is another broadly used term that includes things like fiber reinforced plastics, high performance concrete, and porous rock and ice. The former are commonly referred to as "composites."
The ultimate goal of multiscale simulation is to straddle the ever-moving line between simulation accuracy and computational efficiency. For this reason, there are a number of different breeds of structural multiscale analysis that attempt to strike a balance.
Multiscale with Analytically-Modeled Microstructure
Before we dive into those, lets explain Multiscale Analysis in a little more detail and why its particularly useful for material design and analysis.
MULTISCALE ANALYSIS OF ADVANCED MATERIALS
Modeling advanced materials accurately is extremely complex because of the number of variables at play. The materials in question are heterogeneous in nature – meaning they have more than one pure constituent, e.g., carbon-fiber + polymer resin or sedimentary rock + gaseous pores. While heterogeneity offers huge advantages in performance (making airplanes, space shuttles and lightweight cars possible), it also introduce difficulties in the engineering design. Presently, there is not enough computational power to include all the important details within a single Finite Element (FE) model – as is customary in industry - because that would require a high-resolution model too complex to be feasibly solved.
It is akin to using a single photograph to try and see planet earth and all of the people inside: it is technologically challenging and very inefficient.
While the simulation and analysis technology for metal structures – like car frames - is quite robust, the analysis of novel “advanced materials” is (for the reasons mentioned above) lagging behind. The consensus is that using conventional techniques (standard FEA), it is not possible to accurately simulate these materials without extensive experimental and empirical 'calibration' data. Thus, the introduction of new materials into a structure results in increased time-to-market and costs.
The state-of-the-art solution is to use Multiscale FEA to divide-and-conquer the problem. To accomplish this, a local scale model (of the material microstructure) is “embedded” within the global scale FE model (of the part). These different scales are analyzed simultaneously and the behavior of one level affects the behavior of the other.
Solving each scale individually and linking their results is several orders of magnitude faster than trying to solve a single high-resolution model containing all relevant details. However, not all multiscale techniques are built alike.
ANALYTICALLY MODELED MICROSTRUCTURES
One technique used to account for microstructural nuances is to use an anaytical equation to model behavior. These equations are developed empirically, by witnessing controlled experiments and generating a relationship between all relevant variables that matches the observed outcomes.
“Semi-analytical methods can be defined as direct micromacro procedures for which the local constitutive equations and criteria are evaluated at the local scale using explicit relations to establish the link between the macroscopic and the microscopic fields. Such method correlates the overall macroscopic behaviour with microstructural responses.”
Some of these techniques aim to homogenize the properties of the local scale; others attempt to capture nonlinear behavior via curve fitting and progressive damage approaches. Many of the most famous techniques - like those evaluated in the World Wide Failure Excercises - are related to the analysis of unidirectional composites. There are others that relate to sphereical or oblong inclusions. The key is that the user must be very aware of the assumptions and bounds of their model when employing one of these techniques.
These methods are certainly more accurate than their single-scale, isotropic predecessors, but fall short when trying to analyze novel parts/materials for which their is not historical correlaries or empirical guide-posts.
FINITE ELEMENT SQUARED (FE2)
FE2, introduced by Feyel in 1998, consists of describing the behavior of heterogeneous structures by using finite element models at both the global and local length scales. At each integration point at the macroscopic scale a representative volume element (RVE) is assigned and a separate finite element computation is performed simultaneously. The macroscopic behavior is thus deduced from the non-linearities in the behavior of the associated microstructure. The model is built up using three main ingredients:
- A modeling of the mechanical behavior at the lower scale (the RVE)
- A localisation rule which determines the local solutions inside the unit cell, for any given overall strain
- A homogenisation rule giving the macroscopic stress tensor, knowing the micromechanical stress state
Because the microstructure is a Finite Element mesh, all standard analysis techniques, failure theories, constitutive models, etc. can be easily applied. For this reason, FE2 is known to be flexible and highly accurate, but it is often regarded as too expensive to be practical in full component designs and optimizations. It cases where non-linearity is modeled,it is as if you are "squaring' the number of computations required for standard FEA.
As computational resources become cheaper,this method will become more commonplace; but that point is likely 5-10 years away. For this reason, there currently are no commercially available softwares that utilize pure FE2.
In our humble opinion, this is the "Goldilocks" of multiscale.
TRUE multiscale in an approach developed and exclusively utilized at MultiMech. It combines the accuracy of FE2 with the computational efficiency of analytical methods. It accomplishes this using a breakthrough approach to computing the response of a microstructural RVE. This technique has been shown to be as, if not more accurate than FE2, and at least 6x less computationally intensive.
Just as in FE2, this approach allows one to explicitly model microstructural details, capture local progressive damage, and the show the coalescence of local scale phenomena into global scale phenomena. In other words, the approach naturally inherits the same amount of flexibility and accuracy found in the widely accepted FEA and FE2.
With this approach, engineers are able to perform component and sub-component designs with production-quality run-times, and can even perform optimization studies with tools like HyperStudy.
The full details of this approach are beyond the scope of this post, but feel free to reach out for more info.
For more information on the different breeds of Multiscale Modeling used for the analysis of composites, check out the great articles below.