2:00pm - 2:30pm
Multiscale modelling and numerical simulations of plant tissue growth
1Heriot-Watt University, United Kingdom; 2ENS de Lyon, France; 3Ecole Polytechnique, Paris, France
In this talk we consider derivation of microscopic model for plant tissue growth. We shall assume that growth depends on the turgor pressure inside plant cells and on mechanical properties and deformation of cell walls and middle lamella. Using multiscale analysis techniques we derive macroscopic model for mechanical deformations and growth of plant tissues. For numerical simulations of the macroscopic model a coupled two-scale numerical simulation scheme is developed and implemented in FreeFEM using the Finite Element Method.
Comparison between numerical simulations for microscopic model, defined of the scale of a single cell, and two-scale numerical simulations for the macroscopic problem, defined on the scale of a plant tissue, provides validation of the macroscopic model. Numerical calculations of the effective macroscopic elasticity tensor are compared with the corresponding analytical approximations.
The two-scale numerical simulations for the macroscopic model are then used to investigate the impact of the heterogeneity in the turgor pressure and/or in mechanical properties of cell walls and middle lamella on the growth of plant tissues, as well as to analyse the difference between stress- and strain-driven growth.
2:30pm - 3:00pm
Multiscale Capable Nodal Integral Method
University of Geneva, Switzerland
The numerical solution of multiscale problems is an active area of research, and there exist several schemes to handle such situations. We are interested here to investigate if a powerful semi-analytical coarse mesh scheme called the "Nodal integral method - NIM" can have multiscale capabilities. In NIM, with the transverse integration process (TIP), the PDE is averaged over a cell in each independent direction. TIP converts the PDE into a set of approximate ODEs, one for each independent variable. The final scheme is obtained using the analytical solution of these ODEs. This analytical pre-processing drastically improves the accuracy of NIM. In general, the analytical solution of variable coefficient ODEs is not possible. While obtaining the analytical solution, variable coefficients are treated as constant over the nodes in NIM; this approximation degrades the quality of the solution and makes it unfit for multiscale problems. To remove this problem, we propose an improved version of NIM capable of solving multiscale problems. In the new approach, we first find the numerical solution over a cell with a very fine mesh using any standard numerical scheme such as FVM, which can be treated as an exact solution. The numerical scheme is then developed using this exact solution along with proper transmission conditions. We found that this approach is very much capable of resolving multiscale problems, which is evident from numerical results. We observe that, when the coefficients have matching slope in two consecutive cells, the proposed method needs only five grid points with a simple 2 point centered difference transmition condition. However, when coefficients are discontinuous for two consecutive nodes, transmission condition with an average slope at the interface is used to obtain the solution.
3:00pm - 3:30pm
Improved multiscale finite element methods for advection-diffusion problems
1CERMICS, Ecole Nationale des Ponts et Chaussées, France; 2MATHERIALS project-team, Inria, France
The multiscale finite element method (MsFEM) is a finite element (FE) approach that allows to solve partial differential equations (PDEs) with highly oscillatory coefficients on a coarse mesh, i.e. a mesh with elements of size much larger than the characteristic scale of the heterogeneities . To do so, MsFEMs use pre-computed basis functions, adapted to the differential operator, thereby taking into account the small scales of the problem.
When the PDE contains dominating advection terms, naive FE approximations lead to spurious oscillations, even in the absence of oscillatory coefficients. Stabilization techniques are to be adopted. In spite of various proposals to combine multiscale and stabilization methods, a universally best method has not yet been identified .
In this contribution, we will shed light on the shortcomings of some previously designed MsFEMs for advection-dominated problems and propose the addition of suitable bubble functions to the approximation space. In particular, we introduce an adaptation of the Crouzeix-Raviart FE in the MsFEM-spirit , for which the stabilization turns out to be more effective than for other MsFEM variants.
The work described in this communication is partly joint work with Alexei Lozinski (Université de Besançon), Frédéric Legoll and Claude Le Bris (Ecole des Ponts and Inria). The support of DIM Math INNOV and Inria is gratefully acknowledged.
 Y. Efendiev and T. Hou. Multiscale Finite Element Methods. Springer-Verlag NewYork, 2009.
 C. Le Bris, F. Legoll, and A. Lozinski. MsFEM à la Crouzeix-Raviart for Highly Oscillatory Elliptic Problems. Chinese Annals of Mathematics, Series B, 34(1):113-138, 2013.
 C. Le Bris, F. Legoll, and F. Madiot. A numerical comparison of some multiscale finite element approaches for advection-dominated problems in heterogeneous media. ESAIM: M2AN, 51(3):851-888, 2017.
3:30pm - 4:00pm
Performance of Stage-Parallel Implicit Runge-Kutta Methods with Fast Multigrid Methods
University of Augsburg, Germany
In my talk, I will present results on implementations of fully stage-parallel solvers for Radau-IIA implicit Runge-Kutta (IRK) time integrators. The application context is partial differential equations discretized by higher-order finite element methods, run on large-scale parallel computers where partitioning only the spatial domain might not offer enough parallelism to saturate the available computing resources. My talk will present key algorithms for the parallel execution of IRK methods, including the use of tensor product formulations to expose parallel operations. Two variants will be considered, one approximating the inverse of the Runge-Kutta matrix by a lower-triangular matrix as well as a direct diagonalization of the Runge-Kutta matrix. While the latter involve fewer solver iterations, the use of complex arithmetic reduces the available parallelism. The opportunities and limitations of these methods using a set of different implementation options against traditional time integrators will be assessed for the model problem of the heat equation. My work relies on recent contributions to the widely used deal.II finite element library for cutting-edge ingredients for the spatial part, including multigrid preconditioners and matrix-free operator evaluation for optimal node-level performance.