Giordano Tierra

On linear schemes for a Cahn-Hilliard diffuse interface model

Numerical schemes to approximate the Cahn-Hilliard equation have been widely studied in recent times due to its connection with many physically motivated problems. In this work we propose two type of linear schemes based on different ways to approximate the double-well potential term. The first idea developed in the paper allows us to design a linear numerical scheme which is optimal from the numerical dissipation point of view meanwhile the second one allows us to design unconditionally energy-stable linear schemes (for a modified energy). We present first and second order in time linear schemes to approximate the CH problem, detailing their advantages over other linear schemes that have been previously introduced in the literature. Furthermore, we compare all the schemes through several computational experiments.

Second order schemes and time-step adaptivity for Allen-Cahn and Cahn-Hilliard models

Abstract In this paper, we focus on efficient second-order in time approximations of the Allen–Cahn and Cahn–Hilliard equations. First of all, we present the equations, generic second-order schemes (based on a mid-point approximation of the diffusion term) and some schemes already introduced in the literature. Then, we propose new ways of deriving second-order in time approximations of the potential term (starting from the main schemes introduced in Guillen-Gonzalez and Tierra (2013)), yielding to new second-order schemes. For these schemes and other second-order schemes previously introduced in the literature, we study the constraints on the physical and discrete parameters that can appear to assure the energy-stability, unique solvability and, in the case of nonlinear schemes, the convergence of Newton’s method to the nonlinear schemes. Moreover, in order to save computational cost we have developed a new adaptive time-stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step. Finally, we compare the behaviour of the schemes and the effectiveness of the adaptive time-stepping algorithm through several computational experiments.

Type IV pili interactions promote intercellular association and moderate swarming of Pseudomonas aeruginosa

Significance The opportunistic pathogen Pseudomonas aeruginosa utilizes both its flagellum and type IV pili (TFP) to facilitate motility, attachment, and colonization. Surface motility such as swarming is thought to precede biofilm formation during infection. We combined laboratory and computational methods to probe the physical interactions of TFP during flagellar-mediated swarming and found that TFP of one cell strongly interact with TFP of other cells, which limits swarming expansion rate. Hence, wild-type P. aeruginosa use cell−cell physical interactions via their TFP to control self-organization within motile swarms. This collective mechanism of cell−cell coordination using TFP allows for moderation of swarming direction of individual cells and avoidance of a toxic environment. Pseudomonas aeruginosa is a ubiquitous bacterium that survives in many environments, including as an acute and chronic pathogen in humans. Substantial evidence shows that P. aeruginosa behavior is affected by its motility, and appendages known as flagella and type IV pili (TFP) are known to confer such motility. The role these appendages play when not facilitating motility or attachment, however, is unclear. Here we discern a passive intercellular role of TFP during flagellar-mediated swarming of P. aeruginosa that does not require TFP extension or retraction. We studied swarming at the cellular level using a combination of laboratory experiments and computational simulations to explain the resultant patterns of cells imaged from in vitro swarms. Namely, we used a computational model to simulate swarming and to probe for individual cell behavior that cannot currently be otherwise measured. Our simulations showed that TFP of swarming P. aeruginosa should be distributed all over the cell and that TFP−TFP interactions between cells should be a dominant mechanism that promotes cell−cell interaction, limits lone cell movement, and slows swarm expansion. This predicted physical mechanism involving TFP was confirmed in vitro using pairwise mixtures of strains with and without TFP where cells without TFP separate from cells with TFP. While TFP slow swarm expansion, we show in vitro that TFP help alter collective motion to avoid toxic compounds such as the antibiotic carbenicillin. Thus, TFP physically affect P. aeruginosa swarming by actively promoting cell−cell association and directional collective motion within motile groups to aid their survival.

Multicomponent model of deformation and detachment of a biofilm under fluid flow

A novel biofilm model is described which systemically couples bacteria, extracellular polymeric substances (EPS) and solvent phases in biofilm. This enables the study of contributions of rheology of individual phases to deformation of biofilm in response to fluid flow as well as interactions between different phases. The model, which is based on first and second laws of thermodynamics, is derived using an energetic variational approach and phase-field method. Phase-field coupling is used to model structural changes of a biofilm. A newly developed unconditionally energy-stable numerical splitting scheme is implemented for computing the numerical solution of the model efficiently. Model simulations predict biofilm cohesive failure for the flow velocity between and m s−1 which is consistent with experiments. Simulations predict biofilm deformation resulting in the formation of streamers for EPS exhibiting a viscous-dominated mechanical response and the viscosity of EPS being less than . Higher EPS viscosity provides biofilm with greater resistance to deformation and to removal by the flow. Moreover, simulations show that higher EPS elasticity yields the formation of streamers with complex geometries that are more prone to detachment. These model predictions are shown to be in qualitative agreement with experimental observations.

Analysis of an augmented mixed-FEM for the Navier-Stokes problem

In this paper we propose and analyze a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a “nonlinear-pseudostress” tensor linking the pseudostress tensor with the convective term, which leads to a mixed formulation with the nonlinearpseudostress tensor and the velocity as the main unknowns of the system. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. The resulting mixed formulation is augmented by introducing Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equation and from the Dirichlet boundary condition, which are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed. Then, the classical Banachs fixed point Theorem and Lax-Milgrams Lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish wellposedness and the corresponding Ceas estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the nonlinear-pseudostress tensor, and continuous piecewise polynomial elements of degree k+1 for the velocity, which leads to an optimal convergent scheme. In addition, we provide two iterative methods to solve the corresponding nonlinear system of equations and analyze their convergence. Finally, several numerical results illustrating the good performance of the method are provided. This contribution is based on joint work with Ricardo Oyarzua (Universidad del Bio-Bio) and Giordano Tierra (Charles University).

An Augmented Mixed Finite Element Method for the Navier--Stokes Equations with Variable Viscosity

A new mixed variational formulation for the Navier--Stokes equations with constant density and variable viscosity depending nonlinearly on the gradient of velocity, is proposed and analyzed here. Our approach employs a technique previously applied to the stationary Boussinesq problem and to the Navier--Stokes equations with constant viscosity, which consists firstly of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, and the pressure. Next, by using an equivalent statement suggested by the incompressibility condition, the pressure is eliminated, and in order to handle the nonlinear viscosity, the gradient of velocity is incorporated as an auxiliary unknown. Furthermore, since the convective term forces the velocity to live in a smaller space than usual, we overcome this difficulty by augmenting the variational formulation with suitable Galerkin-type terms arising from the constitutive and equilibrium equations, the aforementioned relation defining the additi...

A posteriori error analysis of an augmented mixed method for the Navier-Stokes equations with nonlinear viscosity

Abstract In this work we develop the a posteriori error analysis of an augmented mixed finite element method for the 2D and 3D versions of the Navier–Stokes equations when the viscosity depends nonlinearly on the module of the velocity gradient. Two different reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary (convex or non-convex) polygonal and polyhedral regions are derived. Our analysis of reliability of the proposed estimators draws mainly upon the global inf–sup condition satisfied by a suitable linearisation of the continuous formulation, an application of Helmholtz decomposition, and the local approximation properties of the Raviart–Thomas and Clement interpolation operators. In addition, differently from previous approaches for augmented mixed formulations, the boundedness of the Clement operator plays now an interesting role in the reliability estimate. On the other hand, inverse and discrete inequalities, and the localisation technique based on triangle-bubble and edge-bubble functions are utilised to show their efficiency. Finally, several numerical results are provided to illustrate the good performance of the augmented mixed method, to confirm the aforementioned properties of the a posteriori error estimators, and to show the behaviour of the associated adaptive algorithm.

A mixed finite element method for Darcy’s equations with pressure dependent porosity

In this work we develop the a priori and a posteriori error analyses of a mixed finite element method for Darcy’s equations with porosity depending exponentially on the pressure. A simple change of variable for this unknown allows to transform the original nonlinear problem into a linear one whose dual-mixed variational formulation falls into the frameworks of the generalized linear saddle point problems and the fixed point equations satisfied by an affine mapping. According to the latter, we are able to show the well-posedness of both the continuous and discrete schemes, as well as the associated Cea estimate, by simply applying a suitable combination of the classical Babuška-Brezzi theory and the Banach fixed point Theorem. In particular, given any integer k ≥ 0, the stability of the Galerkin scheme is guaranteed by employing Raviart-Thomas elements of order k for the velocity, piecewise polynomials of degree k for the pressure, and continuous piecewise polynomials of degree k+1 for an additional Lagrange multiplier given by the trace of the pressure on the Neumann boundary. Note that the two ways of writing the continuous formulation suggest accordingly two different methods for solving the discrete schemes. Next, we derive a reliable and efficient residualbased a posteriori error estimator for this problem. The global inf-sup condition satisfied by the continuous formulation, Helmholtz decompositions, and the local approximation properties of the Raviart-Thomas and Clément interpolation operators are the main tools for proving the reliability. In turn, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of both methods, confirming the aforementioned properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are reported.

Superconvergence in velocity and pressure for the 3D time-dependent Navier-Stokes equations

This work is devoted to the superconvergence in space approximation of a fully discrete scheme for the incompressible time-dependent Navier-Stokes Equations in three-dimensional domains. We discrete by Inf-Sup-stable Finite Element in space and by a semi-implicit backward Euler (linear) scheme in time.Using an extension of the duality argument in negative-norm for elliptic linear problems (see for instance [1]) to the mixed velocity-pressure formulation of the Stokes problem, we prove some superconvergence in space results for the velocity with respect to the energy-norm, and for a weaker norm of L2(0, T; L2(Ω)) type (this latter holds only for the case of Taylor-Hood approximation). On the other hand, we also obtain optimal error estimates for the pressure without imposing constraints on the time and spatial discrete parameters, arriving at superconvergence in the H1 (Ω)-norm again for Taylor-Hood approximations. These results are numerically verified by several computational experiments, where two splitting in time schemes are also considered.

Unconditionally energy stable numerical schemes for phase-field vesicle membrane model

Abstract Numerical schemes to simulate the deformation of vesicles membranes via minimizing the bending energy have been widely studied in recent times due to its connection with many biological motivated problems. In this work we propose a new unconditionally energy stable numerical scheme for a vesicle membrane model that satisfies exactly the conservation of volume constraint and penalizes the surface area constraint. Moreover, we extend these ideas to present an unconditionally energy stable splitting scheme decoupling the interaction of the vesicle with a surrounding fluid. Finally, the well behavior of the proposed schemes are illustrated through several computational experiments.