Multiscale Computations Made Easy

This post on Innovation Intelligence is written by Dr. Jacob Fish, an expert in multiscale computations, professor at Columbia University and founder of Multiscale Design Systems, LLC (MDS). MDS is a member of the Altair Partner Alliance.

If you are an academic and your expertise is in scientific computing, then pursuing multiscale computations would be extremely rewarding. On May 21, 2014, a Google search of a keyword “multiscale” resulted in 1.77 million hits, mostly from the academic community. Twelve faculty positions in multiscale computations were advertised, four journals included the word multiscale in their title, and one third of all the talks in the US National Congress in Computational Mechanics (USNCCM) focused on multiscale computations in 2013 alone. In academia, multiscale computations are taking a center stage in material science, turbulence, weather predictions, biology and finance. Consider for instance the fast growing field of composites, which comprises 7.25% of all the structural materials. The figure below depicts various segments of global composite markets. Projections indicate that in 2016, the composite market will exceed $27 billion (with the end market being considerably larger). Yet, the total revenues produced by all multiscale software development companies today including e-Xstream (MSC), Firehole, AlphaStar, Engenuity, MDS and others are estimated to be less than 0.1% of the global composite market.

Global composite materials shipment by market segment

It turns out that despite prevailing enthusiasm about multiscale computations in the academic community, the private sector remains skeptical. The figure below depicts the building block certification approach pursued in the aerospace industry. The CMH-17 composite materials handbook describes a detailed experimental program to be pursued in the certification process of components made of composite materials.

building block approach

Building block approach from Composite Materials Handbook CMH-17

Typically, as one proceeds up the building blocks, the number of testing samples declines as structural complexity increases. For example, the greatest number of specimens is usually tested at the lowest level when factors such as batch variability and environment are included in the test. The number of specimens approaches 1000 mechanical tests at this lower level. In contrast, only one test article is used at the highest level of the building block, which usually consists of a full-scale test.  As the complexity of higher-level testing increases, more variables are introduced, i.e., fastener holes and multidirectional laminates.

This finding poses an obvious question: why are multiscale methods rarely employed in practice to reduce testing? The prevailing answer is that despite progress made, multiscale methods need to overcome several key hurdles before they will be widely adopted:


  1. Computational complexity barrier – The computational complexity of solving just a two-scale nonlinear problem is tremendous. To illustrate, consider a macro-problem with  Gauss points,  load increments in the macroscale, and  and  average iterations in the macro- and micro- scales, respectively. The total number of linear solves of a micro-problem is  – a formidable computational cost if the number of unit cells and degrees-of-freedom in a unit cell are substantial.
  2. Scale coupling barrier – In metals, material scale, such as grain size in a polycrystaline material, is significantly smaller than structural features. In woven composites, on the other hand, the unit cell size is often of the same order of magnitude as that of the component, like holes and cutouts. Therefore, scales may no longer be separable, rendering homogenization theory questionable.
  3. Existence of defects barrier – While small random variations in composites are unlikely to significantly affect linear properties, they are critical in predicting evolution of failure with any confidence.


Based on more than 30 years of research on the multiscale method, Prof. Jacob Fish of Columbia University, together with his associates, developed scale-separation-free stochastic reduced order multiscale method and, consequently, founded a software development company, Multiscale Design Systems (MDS). The salient features of MDS are as follows:


a)      MDS is equipped with a systematic model reduction technology that reduces complex unit cells having hundreds of thousands of finite elements to a manageable number of deformation modes and state variables.

b)      MDS is free of scale separation and provides mesh insensitive results. The characteristic material length scale is identified based on experimental data.

c)      MDS is equipped with stochastic multiscale capabilities that translate geometrical and material uncertainties into the component level uncertainties.


For more details about MDS theory, please consult the multiscale textbook: J. Fish, Practical Multiscaling, Wiley 2013.

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