Recent and Ongoing Projects

Closed unstretchable knotless ribbons

For the past half-century, efforts to find the equilibrium shapes of Möbius bands and other closed ribbons made from essentially unstretchable materials like paper have relied almost exclusively on numerical strategies for minimizing a dimensionally reduced bending-energy functional provided by Wunderlich. In work done with Yi-chao Chen and Eliot Fried, we focused on filling substantial gaps left in Wunderlich's derivation. The dimensional reduction rests on the fact that the closed ribbon lies on a developable surface which is a ruled surface and, hence, can be parameterized by two vector functions. Our work focused on (i) the constraints that the vector functions must satisfy to ensure that the underlying deformation from a flat rectangular strip to the ribbon is an isometric immersion, (ii) the boundary conditions and corresponding requirements on the self-linking number of the midline of the band that determine whether or not the band is orientable and how many twists it possesses, (iii) showing that if the two vector functions satisfy the constraints then they can be used to construct a surface and an isometric deformation from a flat rectangular strip to that surface and (iv) the derivation of the complete set of Euler–Lagrange equations for an augmented energy functional that properly incorporates Lagrange and Karush–Kuhn–Tucker multipliers associated with the constraints. Our analysis allows for the possibility that rulings may meet at isolated points on the edge of a band but not in its interior.

Nonlocal curvature

Roberto Paroni, Paolo Podio-Guidugli, and I extended the definition of nonlocal mean-curvature to any smooth hypersurface when previously this notion only existed for surfaces that are the boundary of an open set. The concept of nonlocal curvature originates in the work of Caffarelli, Roquejoffre, and Savin. They studied the minimizers of a fractional perimeter functional related to the fractional Sobolev-norm of the characteristic function of an open set. In the case of a bounded, open set $E\subset\mathbb{R}^n$, the $s$-perimeter, for $s\in (0,1)$, is given by \begin{equation}\label{Pers} \text{Per}_s(E)=\frac{1}{\alpha_{n-1}} \int_E\int_{E^c}|x-y|^{-n-s}dxdy, \end{equation} where $\alpha_{n-1}$ is the volume of the unit ball in $\mathbb{R}^{n-1}$. The sets that minimize the $s$-perimeter functional subject to a boundary-like condition were found to satisfy a pointwise condition on their boundary which was used to define a nonlocal mean-curvature. The relationship between this nonlocal curvature and the classical mean-curvature was explored by Abatangelo and Valdinoci. To extend the idea of nonlocal mean-curvature to any smooth surface, first my collaborators and I showed that for bounded $E$ with smooth boundary \begin{equation}\label{PersR} \text{Per}_s(E)=\frac{1}{2\alpha_{n-1}}\int_{{\cal X}(\partial E)}|x-y|^{-n-s}dxdy, \end{equation} where ${\cal X}(\partial E)$ is the set of all pairs of points $(x,y)\in\mathbb{R}^n\times\mathbb{R}^n$ such that the oriented line segment connecting $x$ to $y$ crosses $\partial E$ an odd number of times. The validity of this follows from the fact that ${\cal X}(\partial E)$ and $(E\times E^c)\cup (E^c\times E)$ agree up to a set of ${\cal H}^{2n}$-measure zero. This second expression for the fractional perimeter allowed us to define a fractional notion of area for a bounded, smooth surface, even when it is not the boundary of a set. We showed that the fractional area of a surface converges to the classical notion of area in an appropriate limit. Moreover, minimizers of the fractional area must satisfy a pointwise condition that we used to define a nonlocal mean-curvature of the surface which converges to the classical curvature in an appropriate limit. In a followup work, I defined a fractional notion of arc length and nonlocal curvature for a curve in $\mathbb{R}^n$. The fractional length is defined through an integral, similar to the second expression for $\text{Per}_s$, except one integrates over the collection of $(n-1)$-dimensional discs that intersect the curve an odd number of times and the integrand involves a power of the radius of the disc. The motivation for this definition is that in two dimensions, when the concepts of a surface and a curve coincide, an argument shows that this definition is equivalent to the definition of the fractional area up to a multiplicative factor. It is shown that in an appropriate limit the fractional length converges to the classical notion of length and the Euler–Lagrange equation of this functional motivates the definition of the nonlocal curvature of a curve. Computing the first variation of the fractional length functional was challenging because the domain of integration depended on the variational parameter in a nontrivial way. Overcoming this difficulty resulted in a separate paper where, using tools from geometric measure theory, I established a general first variation formula that may have applications outside of the area of nonlocal curvature.

Multi-component multiphase flow

Noel Walkington and I formulated an axiomatic foundation for the models of multi-component multiphase porous flow appearing ubiquitously in the engineering literature. tions of porous flow formulate the problem in the context of mixture theory where each constituent is treated as a separate continuum with its own motion and balance laws. This results in a large system of coupled PDEs which are impractical to solve numerically. To circumvent this, motivated by results in homogenization theory, we replaced the PDEs coming from the balance of foces with Darcy's laws, which are algebraic equations for the velocity of each phase. We also assumed that thermodynamic equilibrium occurs on a time scale much shorter than the pore-to-pore transport. This allowed for the use of classical thermodynamics to determine the composition and volume fraction of each phase within the pores. To ensure that our proposed constitutive laws were consistent with the second law of thermodynamics, the Coleman–Noll procedure was used. Moreover, the corresponding dissipation inequalities establish stability of solutions. The convexity properties and variational structure of our proposed model was also investigated. The above analysis was done for a flow through a rigid porous medium. However, there are theories of porous flow in which the medium is assumed to be linearly elastic, such as in Biot's theory, but these theories only involve a single fluid phase. Expanding on our previous work, Noel Walkington and I allowed for the porous medium to deform elastically. Among other things, we showed the Biot's theory is a special case of ours with the Biot parameters appearing as a result of a linearization.

Publications



Seguin, B., Chen, Y.C., Fried, E.: Closed unstretchable knotless ribbons and the Wunderlich functional. submitted

Zambom, A.Z., Seguin, B.: Fastest route planning for an unmanned vehicle in the presence of accelerating obstacles. submitted

Seguin, B., Walkington N.J.: Multi-component multiphase flow. accepted at Archive for Rational Mechanics and Analysis

Seguin, B.: A Fractional Notion of Length and an Associated Nonlocal Curvature. The Journal of Geometric Analysis [link] [ArXiv]

Seguin, B.: A transport theorem for nonconvecting open sets on an embedded manifold. Continuum Mechanics and Thermodynamics [link] [ArXiv]

Seguin, B., Walkington N.J.: Multi-component multiphase flow through a poroelastic medium. Journal of Elasticity 135, 485–507 (2019) [link]

Zambom, A., Seguin, B., Zhao, F.: Robot path planning in a dynamic environment with stochastic measurements. Journal of Global Optimization 73, 389–410 (2019) [link]

Paroni, R., Podio-Guidugli, P., Seguin, B.: On the nonlocal curvatures of open surfaces with or without boundary. Communications of Pure and Applied Analysis 17, 709–727 (2018) [link] [ArXiv]

Seguin, B.: On the homogenization of a new class of locally periodic microstructures in linear elasticity with residual stress. Mathematics and Mechanics of Solids 23, 1025–1039 (2017) [link] [ArXiv]

Ptashnyk, M., Seguin, B.: Homogenization of a viscoelastic model for plant cell wall biomechanics. ESIAM: Control, Optimisation and Calculus of Variations 23, 1447–1471 (2017) [link] [ArXiv]

Ptashnyk, M., Seguin, B.: The impact of microfibril orientations on the biomechanics of plant cell walls and tissues: modelling and simulations. Bulletin of Mathematical Biology 78, 2135–2164 (2016) [link]

Ptashnyk, M., Seguin, B.: Periodic homogenization and material symmetry in linear elasticity. Journal of Elasticity 124, 225–241 (2016) [link]

Ptashnyk, M., Seguin, B.: Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics. ESIAM Mathematical Modelling and Numerical Analysis 50, 593–631 (2016) [link] [ArXiv]

Seguin, B., Fried, E.: Stable and unstable helices: Soap films in cylindrical tubes. Calculus of Variations and Partial Differential Equations 54, 969–988 (2015) [link] [ArXiv]

Seguin, B., Fried, E.: Calculating the bending moduli of the Canham–Helfrich free-energy density from a particular potential. ICMS Workshop: Differential Geometry and Continuum Mechanics. Proceedings in Mathematics and Statistics. Springer (2015) [ArXiv]

Seguin, B., Hinz, D. F., Fried, E.: Extending the Transport Theorem to Rough Domains of Integration. Applied Mechanics Reviews 66, 050802 (2014) [link]

Seguin, B., Fried, E.: Roughening it — Evolving irregular domains and transport theorems. Mathematical Models and Methods in Applied Sciences 24, 1729 (2014) [link]

Seguin, B., Fried, E.: Microphysical derivation of the Canham–Helfrich free-energy density. Journal of Mathematical Biology 68, 647–665 (2014) [link] [ArXiv]

Maleki, M., Seguin, B., Fried, E.: Kinematics, material symmetry, and energy densities for lipid bilayers with spontaneous curvature. Biomechanics and Modeling in Mechanobiology 12, 997–1017 (2013) [link]

Seguin, B., Fried, E.: Statistical foundations of liquid-crystal Theory II. Continuum-level balances. Archive for Rational Mechanics and Analysis 207, 1–37 (2013) [link]

Seguin, B., Fried, E.: Statistical foundations of liquid-crystal Theory I. Discrete systems of rod-like molecules. Archive for Rational Mechanics and Analysis 206, 1039–1072 (2012) [link]

Capriz, G., Fried, E., Seguin, B.: Constrained ephemeral continua. Rendiconti Lincei–Matematica e Applicazioni 23, 157–195 (2012) [link]

Seguin, B.: Simple thermomechanical materials with memory. Journal of Elasticity 105, 207–252 (2011) [link]

Seguin, B.: Thermoelasto-viscous materials. Journal of Elasticity 101, 153–177 (2010) [link]

Noll, W., Seguin, B.: Basic concepts in thermomechanics. Journal of Elasticity 101, 121–151 (2010) [link]

Noll, W., Seguin, B.: Plugs in viscometric flows of simple semi-liquids. Journal of the Society of Rheology Japan 37, 1–10 (2009) [link]

Noll, W., Seguin, B.: Monoids, boolean algebras, and materially ordered sets. International Journal of Pure and Applied Mathematics 37, 187–202 (2007) [link]