Phase Fields and Computational Mechanics

Thomas J.R. Hughes

Oden Institute, University of Texas at Austin

To many of us, a perhaps overly simplified description of a Computational Mechanics method is one composed of a variational method and function spaces. It is even simplified further if we assume a Galerkin variational method, which only entails one function space. That is a paradigm that has had enormous practical benefits, being the basis of much large-scale computing done in engineering and science. Usually, this can be phrased in terms of the weak form of the problem, the starting point of discretization. It is the discretization of the function spaces that we deal with in practice, reducing an infinite dimensional problem to a finite dimensional one that can be solved on a computer. The Finite Element Method is obviously the predominant discretization method, and the first one to combine geometric and topologic versatility. In a sense, no matter how complicated an engineering design, it can be discretized using the Finite Element Method. However, it is well known that the exact geometry, perhaps defined by a Computer Aided Design (CAD) file, is almost never represented exactly by the Finite Element Method, the only exceptions being very simple cases. This is but one deficiency of the contemporary Finite Element Method in practice. One can add that building meshes is labor intensive, and a significant bottleneck in the design-through-analysis process. Other deficiencies are the introduction of geometry errors in computational models that arise due to feature removal, geometry clean-up and CAD “healing,” utilized to facilitate efficient mesh generation. Still other shortcomings of contemporary technology are the inability to “close the loop” with design optimization, and the lack of robustness of higher-order finite elements to achieve their full promise in industrial applications. What has been done to address these issues?

Isogeometric Analysis [1] in its basic form represents a partial solution. It is based on the geometrical representations used in CAD, predominantly smooth splines, and is capable of more precise geometric descriptions, and more robust performance of higher-order spline elements, compared with standard higher-order C⁰-continuous finite elements, but the problem of developing boundary-fitted meshes remains laborious. Shortly after the introduction of Isogeometric Analysis, Ernst Rank and Alexander Düster proposed the Finite Cell Method, a cut-element or immersed method. In contrast with classical immersed methods, they showed how to obtain the same accuracy as the boundary-fitted method, and specifically higher-order accuracy with higher-order elements. There initial work was for standard higher-order elements but they soon after applied it to Isogeometic Analysis. It has been subsequently shown that Isogeometric Analysis has analytical advantages over standard finite elements in the immersed setting. This has facilitated the original dream of Isogeometric Analysis: To create exact geometries, expedite mesh generation and simplify local refinement. It seems the key concept is the introduction of a phase field that defines the geometry. In the case of an engineering design, the CAD file suffices to defines the phase field. It is binary in this instance, taking on the value 1.0 where there is material and 0.0 elsewhere. The phase field concept can be brought to life as a continuous function, which enables the integration of other types of analysis, such as topology optimization within CAD, additive manufacturing, and phase field fracture.

Phase fields are everywhere in contemporary Computational Mechanics and are advocated as a standard device going forward. Immersed and phase field analysis will be illustrated through examples and applications, including Computational Medicine. I also hope to address the open question of whether we can immerse a geometry in an artificial neural net, say a Variationally Mimetic Operator Network (VarMiON), and obtain better, worse or equivalent results to standard Finite Element or Isogeometric Analysis Methods.

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[1] T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs, “Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement,” Computer Methods in Applied Mechanics and Engineering, 194, (2005) 4135-4195.