Recent Significant Accomplishments in Research

Professor Costanzo’s research in computational mechanics has focused on the development of com- putational tools rooted in the physics of the problems. The following publications highlight his most recent major accomplishments in research.

(i) Roy, S., L. Heltai, and F. Costanzo (2015), “Benchmarking the Immersed Finite Element Method for Fluid-Structure Interaction Problems”, Computers and Mathematics with Appli- cations, 69(10), pp. 1167–1188. DOI: 10.1016/j.camwa.2015.03.012.

FSI problems are very challenging to solve numerically and there are not closed-form solutions that can be generated (say, with the method of manufactured solutions) for the purpose of assessing convergence and computational efficiency. In lieu of closed-formed solutions, the FSI community has de facto agreed to use as benchmarks a collection of numerical tests com- piled by Tureck and Hron in 2006. The significance of the above work is that, until now, no immersed method was ever been shown to strictly conform to the Tureck-Hron benchmarks. Costanzo’s group is the first to have produced results conforming to these benchmarks with an immersed method. This accomplishment is extraordinarily important in the FSI community.

(ii) Nama, N., R. Barnkob, Z. Mao, C. J. Kähler, F. Costanzo, and T. J. Huang (2015), “Numerical Study of Acoustophoretic Motion of Particles in a PDMS Microchannel Driven by Surface Acoustic Waves”, Lab on a Chip, in press. DOI: 10.1039/C5LC00231A; PMID: 26001199.

This paper is focused on the prediction of the forces acting on particles suspended in a streaming flow within a microacoustofluidic device. Studies of the effects of bulk waves are common. However, the acoustophoretic properties of flows activated by surface acoustic waves are scarce. This paper represents a first rigorous computational study of such motions. This work fits in a general theme of FSI and is currently being generalized to predict the motion of soft inclusions such as cells in microacoustofluidic devices for the purpose of quantifying their mechanical properties.

(iii) Nama, N., P.-H. Huang, T. J. Huang, and F. Costanzo (2014), “Investigation of Acous- tic Streaming Patterns around Oscillating Sharp Edges”, Lab on a Chip, 14(15), pp. 2824– 2836. DOI: 10.1039/c4lc00191e; PMID: 24903475; PMCID: PMC4096312.

Microacoustofluidics deals with the creation of controlled flow patterns for fluids in channels with micron-sized lumen diameter. The flow is activated by high-frequency piezoelectric actu- ators that “shake” the channel and, along with the primary harmonic response, activate fluid flows that are not harmonic and result from the interplay between the dissipative nature of the fluids and the compressible part of their constitutive response. Slow drifts activated by har- monic excitation of fluid have been studied for a long time, but not at the micro-level. More importantly, this field has not received a careful analytical and numerical treatment leaving a large number of technical difficult problems unsolved. This paper predicts streaming flows in a micro-mixer whose channels have walls with has sharp edges. The key innovation here is a singularity analysis for fluid flows analogous to that of stress fields around crack tips in linear elasticity.

(iv) Costanzo, F. and J. G. Brasseur (2013), “The Invalidity of the Laplace Law for Biological Vessels and of Estimating Elastic Modulus from Total Stress vs. Strain: a New Practical Method”, Mathematical Medicine & Biology, first published online: September 25, 2013, DOI: 10.1093/imammb/dqt020.

Various mechanical tests are performed in vivo in a clinical setting on tubular organs, like the esophagus. Because of the intrinsic limitations of tests done on live human subjects, these tests provided only limited information. More importantly, for various reasons, the information collected in a clinical setting is routinely interpreted via formulas that are appealing to non- mechanicians, but that are compromised by the quasi-incompressibility of the tissue and the presence of residual stress. In this context, the above paper has provided a re-analysis of the inflation and extension tests of circular cylinders and a re-interpretation of the data collected by common tests so to provide new formulas for the mechanistically correct estimation of the effective shear modulus of the tissue. The formulas in question require exactly the same input as current clinical practices. However, they present clinicians with a paradigm shift in which the estimate of the tangent moduli (information that is polluted in a quasi-incompressible context) is now replaced by the correct estimation of the tissue’s shear modulus. The work in this paper has the potential to have an strong impact in clinical practice in that it provides clinicians with a methodology to correctly assess the elastic response of tissues of tubular organs in live patients. Drs. Brasseur and Costanzo are now collaborating on an outreach effort to publicize the new methodology to clinicians.

(v) Heltai, L. and F. Costanzo (2012), “Variational Implementation of Immersed Finite Element Methods”, Computer Methods in Applied Mechanics and Engineering, 229–232, pp. 110–127.

Fluid structure interaction problems in biomedical applications are challenging because one must often model objects that are fully immersed in a fluid and that undergo large displace- ments and rotations in addition to large strains. For example, such is the motion of red blood cells in going from arteries, to arterioles, to capillaries. Immersed methods, originally devel- oped by C. Peskin in the late 1970s for the modeling of blood flow in the hearth, have proven to be far more flexible for biomedical applications than more traditional ALE approaches. Peskin’s approach and later generalizations have suffered from strong limitations, such as the requirement that the immersed body be incompressible with the same density as the surround- ing fluid, the latter also incompressible. The significance of the above paper is the generation of an FEM scheme that is not limited by constitutive assumptions on either the fluid or the solid. Also, it is the first paper that presents a formal proof of the numerical stability of the method. For this reasons the above is a major contribution in the FSI literature.

(vi) Costanzo, F., G. L. Gray, and P. C. Andia (2005) “On the Definitions of Effective Stress and Deformation Gradient for Use in MD: Hill’s Macro-Homogeneity and the Virial Theo- rem,” International Journal of Engineering Science, 43(7), pp. 533–555.

As engineering applications have moved to the submicron- and nano-scales, classically-trained engineers started using tools from statistical mechanics and solid state physics like molecu- lar dynamics. As it turns out, there were no theoretical results available to tie together the averaging notions from homogenization theory of composite materials with the averaging pro- cesses carried out in molecular dynamics simulations. The central contribution of this paper is the construction of a mathematically rigorous bridge between molecular dynamics, as an intrinsically discrete approach to the estimation of material properties, and continuum mechanics. In particular, the authors have established a rigorous limit process for the matching of the classical Cauchy stress and the so-called virial stress.