Finite strain porohyperelasticity: An asymptotic multiscale ALE-FSI approach supported by ANNs

The governing equations and numerical solution strategy to solve porohyperelstic problems as multiscale multiphysics media are provided in this contribution. The problem starts from formulating and non-dimensionalising a Fluid-Solid Interaction (FSI) problem using Arbitrary Lagrangian-Eulerian (ALE) technique at the pore level. The resultant ALE-FSI coupled systems of PDEs are expanded and analysed using the asymptotic homogenisation technique which yields three partially novel systems of PDEs, one governing the macroscopic/effective problem supplied by two microscale problems (fluid and solid). The latter two provide the microscopic response fields whose average value is required in real-time/online form to determine the macroscale response. This is possible efficiently by training an Artificial Neural Network (ANN) as a surrogate for the Direct Numerical Solution (DNS) of the microscale solid problem. The present methodology allows to solve finite strain (multiscale) porohyperelastic problems accurately using direct derivative of the strain energy, for the first time. Furthermore, a simple real-time output density check is introduced to achieve an optimal and reliable training dataset from DNS. A Representative Volume Element (RVE) is adopted which is followed by performing a microscale (RVE) sensitivity analysis and a multiscale confined consolidation simulation showing the importance of employing the present method when dealing with finite strain poroelastic/porohyperelastic problems

ANN-aided incremental multiscale-remodelling-based finite strain poroelasticity

Mechanical modelling of poroelastic media under finite strain is usually carried out via phenomenological models neglecting complex micro-macro scales interdependency. One reason is that the mathematical two-scale analysis is only straightforward assuming infinitesimal strain theory. Exploiting the potential of ANNs for fast and reliable upscaling and localisation procedures, we propose an incremental numerical approach that considers rearrangement of the cell properties based on its current deformation, which leads to the remodelling of the macroscopic model after each time increment. This computational framework is valid for finite strain and large deformation problems while it ensures infinitesimal strain increments within time steps. The full effects of the interdependency between the properties and response of macro and micro scales are considered for the first time providing more accurate predictive analysis of fluid-saturated porous media which is studied via a numerical consolidation example. Furthermore, the (nonlinear) deviation from Darcy's law is captured in fluid filtration numerical analyses. Finally, the brain tissue mechanical response under uniaxial cyclic test is simulated and studied

A hybrid MGA-MSGD ANN training approach for approximate solution of linear elliptic PDEs

We introduce a hybrid "Modified Genetic Algorithm-Multilevel Stochastic Gradient Descent" (MGA-MSGD) training algorithm that considerably improves accuracy and efficiency of solving 3D mechanical problems described, in strong-form, by PDEs via ANNs (Artificial Neural Networks). This presented approach allows the selection of a number of locations of interest at which the state variables are expected to fulfil the governing equations associated with a physical problem. Unlike classical PDE approximation methods such as finite differences or the finite element method, there is no need to establish and reconstruct the physical field quantity throughout the computational domain in order to predict the mechanical response at specific locations of interest. The basic idea of MGA-MSGD is the manipulation of the learnable parameters' components responsible for the error explosion so that we can train the network with relatively larger learning rates which avoids trapping in local minima. The proposed training approach is less sensitive to the learning rate value, training points density and distribution, and the random initial parameters. The distance function to minimise is where we introduce the PDEs including any physical laws and conditions (so-called, Physics Informed ANN). The Genetic algorithm is modified to be suitable for this type of ANN in which a Coarse-level Stochastic Gradient Descent (CSGD) is exploited to make the decision of the offspring qualification. Employing the presented approach, a considerable improvement in both accuracy and efficiency, compared with standard training algorithms such as classical SGD and Adam optimiser, is observed. The local displacement accuracy is studied and ensured by introducing the results of Finite Element Method (FEM) at sufficiently fine mesh as the reference displacements. A slightly more complex problem is solved ensuring its feasibility

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

International audienceThe paper introduces a weighted residual-based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple-flow-immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non-linear kinematics is mapped to the flow using the zero iso-contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi-field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non-smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail

Finite element method for strongly-coupled systems of fluid-structure interaction with application to granular flow in silos

A monolithic approach to fluid-structure interactions based on the space-time finite element method (STFEM) is presented. The method is applied to the investigation of stress states in silos filled with granular material during discharge. The thin-walled siloshell is modeled in a continuum approach as elastic solid material, whereas the flowing granular material is described by an enhanced viscoplastic non-Newtonian fluid model. The weak forms of the governing equations are discretized by STFEM for both solid and fluid domain. To adapt the matching mesh nodes of the fluid domain to the structural deformations, a mesh-moving scheme using a neo-Hookean pseudo-solid is applied. The finite element approximation of non-smooth solution characteristics is enhanced by the extended finite element method (XFEM). The proposed methodology is applied to the 4D (space-time) investigation of deformation-dependent loading conditions during silo discharge

Toward fluid-structure-piezoelectric simulations applied to flow-induced energy harvesters

The subject deals with the simulation of flow-induced energy harvesters. We focus in particular on the modelling of autonomous piezo-ceramic power generators to convert ambient fluid-flow energy into electrical energy. The vibrations of an immersed electromechanical structure with large amplitude have to be taken into account in that case. One challenge consists in modelling and predicting the nonlinear coupled dynamic behaviour for the improved design of such devices. The set of governing equations is expressed in integral form, using the method of weighted residuals, and discretized with finite elements using the open source package FEniCS. Preliminary results of separated problems using FEniCS will be detailed and discussed (e.g. Navier-Stokes with or without moving meshes, nonlinear elasticity, aeroelasticity and electromechanical coupling). The objective is to validate each problem independently before coupling all the phenomena in a monolithic framework. Those simulations involve nonlinearities at many levels of modeling. The perspective of using reduced order models to limit the computational cost (in time and memory) will be discussed in an outlook to this work

Non-localised contact between beams with circular and elliptical cross-sections

The key novelty of this contribution is a dedicated technique to e fficiently determine the distance (gap) function between parallel or almost parallel beams with circular and elliptical cross-sections. The technique consists of parametrizing the surfaces of the two beams in contact, fixing a point on the centroid line of one of the beams and searching for a constrained minimum distance between the surfaces (two variants are investigated). The resulting unilateral (frictionless) contact condition is then enforced with the Penalty method, which introduces compliance to the, otherwise rigid, beams' cross-sections. Two contact integration schemes are considered: the conventional slave-master approach (which is biased as the contact virtual work is only integrated over the slave surface) and the so-called two-half-pass approach (which is unbiased as the contact virtual work is integrated over the two contacting surfaces). Details of the finite element formulation which is suitably implemented using Automatic Di fferentiation techniques are presented. A set of numerical experiments shows the overall performance of the framework and allows a quantitative comparison of the investigated variants

Nonlinear analysis of thin-walled structures based on tangential differential calculus with FEniCSx

We present an approach to implement the Tangential Differential Calculus (TDC) for a variety of thin-walled structures (beams, membranes, shells) in the framework of nonlinear kinematics and/or material behaviour. In contrast to classical formulations the TDC describes kinematics, equilibrium and constitutive relation of the thin structure (as two-dimensional manifold) on the basis of a full three-dimensional deformation state. This allows to introduce the undeformed configuration of e.g. a shell directly in terms of a mesh of topological dimension 2 and geometrical dimension 3. Of particular interest is the use of finite elements of higher-order geometrical order to capture the (interpolated) curvature of the manifold with high accuracy. Numerical examples and reference implementations of this work to support nonlinear stress and post-buckling analyses (using a realisation of the classical arc-length method in FEniCSx) will be provided as a part of the package dolfiny (https://github.com/michalhabera/dolfiny)

Nonlinear local solver

Many engineering applications require solution of a global finite element problem coupled with nonlinear equations of local nature. Local in the sense, that for a known global state the local solution could be found on cell-by-cell basis. Examples include plastic deformation problems, static condensation (hybridization) of displacement-stress formulation or just a simple nonlinear constitutive laws to be satisfied at each quadrature point. These types of problems either required special libraries and extensions in order to be solved with FEniCS (and FEniCS-X) tools, or lead to very slow implementations due to hacks and tricks needed to achieve the solution (e.g. monolithic schemes which increase the matrix problem size). In this talk a unified approach tailored for the current state of FEniCS-X interfaces is presented. The approach computes consistent global tangent operator for nonlinear problems. In addition, local equations are formulated symbolically in UFL, and their derivatives are therefore computed automatically. Several low-level examples (incl. plasticity with symbolic yield surface, nonlinear static condensation and materials with implicit constitutive laws) that demonstrate the main concepts are presented. Finally, high-level wrappers for this functionality are presented. These come as a part of package `dolfiny` (https://github.com/michalhabera/dolfiny)