Seamless Adaptivity of Elastic Models

International audienceA new adaptive model for viscoelastic solids is presented. Unlike previous approaches, it allows seamless transitions, and simplifications in deformed states. The deformation field is generated by a set of physically animated frames. Starting from a fine set of frames and mechanical energy integration points, the model can be coarsened by attaching frames to others, and merging integration points. Since frames can be attached in arbitrary relative positions, simplifications can occur seamlessly in deformed states, without returning to the original shape, which can be recovered later after refinement. We propose a new class of velocity-based simplification criterion based on relative velocities. Integration points can be merged to reduce the computation time even more, and we show how to maintain constant elastic forces through the levels of detail. This meshless adaptivity allows significant improvements of computation time

An adaptive meshfree method for phase-field models of biomembranes. Part II: A Lagrangian approach for membranes in viscous fluids

We present a Lagrangian phase-field method to study the low Reynolds number dynamics of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the field variables are the phase-field and the fluid velocity, here we exploit the fact that the phasefield tracks a material interface to reformulate the problem in terms of the Lagrangian motion of a background medium, containing both the biomembrane and the fluid. We discretize the equations in space with maximum-entropy approximants, carefully shown to perform well in phase-field models of biomembranes in a companion paper. The proposed formulation is variational, lending itself to implicit time-stepping algorithms based on minimization of a time-incremental energy, which are automatically nonlinearly stable. The proposed method deals with two of the major challenges in the numerical treatment of coupled fluid/phase-field models of biomembranes, namely the adaptivity of the grid to resolve the sharp features of the phase-field, and the stiffness of the equations, leading to very small time-steps. In our method, local refinement follows the features of the phasefield as both are advected by the Lagrangian motion, and large time-steps can be robustly chosen in the variational time-stepping algorithm, which also lends itself to time adaptivity. The method is presented in the axisymmetric setting, but it can be directly extended to 3D. We present a Lagrangian phase-field method to study the low Reynolds number dynamics of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the field variables are the phase-field and the fluid velocity, here we exploit the fact that the phase-field tracks a material interface to reformulate the problem in terms of the Lagrangian motion of a background medium, containing both the biomembrane and the fluid. We discretize the equations in space with maximum-entropy approximants, carefully shown to perform well in phase-field models of biomembranes in a companion paper. The proposed formulation is variational, lending itself to implicit time-stepping algorithms based on minimization of a time-incremental energy, which are automatically nonlinearly stable. The proposed method deals with two of the major challenges in the numerical treatment of coupled fluid/phase-field models of biomembranes, namely the adaptivity of the grid to resolve the sharp features of the phase-field, and the stiffness of the equations, leading to very small time-steps. In our method, local refinement follows the features of the phase-field as both are advected by the Lagrangian motion, and large time-steps can be robustly chosen in the variational time-stepping algorithm, which also lends itself to time adaptivity. The method is presented in the axisymmetric setting, but it can be directly extended to 3D

ADAPTIVE REFINEMENT AND MULTISCALE MODELING IN 2D PERIDYNAMICS

The original peridynamics formulation uses a constant nonlocal region, the horizon, over the entire domain. We propose here adaptive refinement algorithms for the bond-based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multiscale modeling, and in peridynamics changing the horizon is directly related to multiscale modeling. We do not use any special conditions for the “coupling” of the large and small horizon regions, in contrast with other multiscale coupling methods like atomistic-to-continuum coupling, which require special conditions at the interface to eliminate ghost forces in equilibrium problems. We formulate, and implement in two dimensions, the peridynamic theory with a variable horizon size and we show convergence results (to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero) for a number of test cases. Our refinement is triggered by the value of the nonlocal strain energy density. We apply the boundary conditions in a manner similar to the way these conditions are enforced in, for example, the finite-element method, only on the nodes on the boundary. This, in addition to the peridynamic material being effectively “softer” near the boundary (the so-called “skin effect”) leads to strain energy concentration zones on the loaded boundaries. Because of this, refinement is also triggered around the loaded boundaries, in contrast to what happens in, for example, adaptive finite-element methods

ADAPTIVE REFINEMENT AND MULTISCALE MODELING IN 2D PERIDYNAMICS

The original peridynamics formulation uses a constant nonlocal region, the horizon, over the entire domain. We propose here adaptive refinement algorithms for the bond-based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multiscale modeling, and in peridynamics changing the horizon is directly related to multiscale modeling. We do not use any special conditions for the “coupling” of the large and small horizon regions, in contrast with other multiscale coupling methods like atomistic-to-continuum coupling, which require special conditions at the interface to eliminate ghost forces in equilibrium problems. We formulate, and implement in two dimensions, the peridynamic theory with a variable horizon size and we show convergence results (to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero) for a number of test cases. Our refinement is triggered by the value of the nonlocal strain energy density. We apply the boundary conditions in a manner similar to the way these conditions are enforced in, for example, the finite-element method, only on the nodes on the boundary. This, in addition to the peridynamic material being effectively “softer” near the boundary (the so-called “skin effect”) leads to strain energy concentration zones on the loaded boundaries. Because of this, refinement is also triggered around the loaded boundaries, in contrast to what happens in, for example, adaptive finite-element methods

Adaptive Mechanisms for Mobile Spatio-Temporal Applications

Mobile spatio-temporal applications play a key role in many mission critical fields, including Business Intelligence, Traffic Management and Disaster Management. They are characterized by high data volume, velocity and large and variable number of mobile users. The design and implementation of these applications should not only consider this variablility, but also support other quality requirements such as performance and cost. In this thesis we propose an architecture for mobile spatio-temporal applications, which enables multiple angles of adaptivity. We also introduce a two-level adaptation mechanism that ensures system performance while facilitating scalability and context-aware adaptivity. We validate the architecture and adaptation mechanisms by implementing a road quality assessment mobile application as a use case and by performing a series of experiments on cloud environment. We show that our proposed architecture can adapt at runtime and maintain service level objectives while offering cost-efficiency and robustness

On-the-fly adaptivity for nonlinear twoscale simulations using artificial neural networks and reduced order modeling

A multi-fidelity surrogate model for highly nonlinear multiscale problems is proposed. It is based on the introduction of two different surrogate models and an adaptive on-the-fly switching. The two concurrent surrogates are built incrementally starting from a moderate set of evaluations of the full order model. Therefore, a reduced order model (ROM) is generated. Using a hybrid ROM-preconditioned FE solver, additional effective stress-strain data is simulated while the number of samples is kept to a moderate level by using a dedicated and physics-guided sampling technique. Machine learning (ML) is subsequently used to build the second surrogate by means of artificial neural networks (ANN). Different ANN architectures are explored and the features used as inputs of the ANN are fine tuned in order to improve the overall quality of the ML model. Additional ANN surrogates for the stress errors are generated. Therefore, conservative design guidelines for error surrogates are presented by adapting the loss functions of the ANN training in pure regression or pure classification settings. The error surrogates can be used as quality indicators in order to adaptively select the appropriate -- i.e. efficient yet accurate -- surrogate. Two strategies for the on-the-fly switching are investigated and a practicable and robust algorithm is proposed that eliminates relevant technical difficulties attributed to model switching. The provided algorithms and ANN design guidelines can easily be adopted for different problem settings and, thereby, they enable generalization of the used machine learning techniques for a wide range of applications. The resulting hybrid surrogate is employed in challenging multilevel FE simulations for a three-phase composite with pseudo-plastic micro-constituents. Numerical examples highlight the performance of the proposed approach

An extended finite element method with smooth nodal stress

The enrichment formulation of double-interpolation finite element method (DFEM) is developed in this paper. DFEM is first proposed by Zheng \emph{et al} (2011) and it requires two stages of interpolation to construct the trial function. The first stage of interpolation is the same as the standard finite element interpolation. Then the interpolation is reproduced by an additional procedure using the nodal values and nodal gradients which are derived from the first stage as interpolants. The re-constructed trial functions are now able to produce continuous nodal gradients, smooth nodal stress without post-processing and higher order basis without increasing the total degrees of freedom. Several benchmark numerical examples are performed to investigate accuracy and efficiency of DFEM and enriched DFEM. When compared with standard FEM, super-convergence rate and better accuracy are obtained by DFEM. For the numerical simulation of crack propagation, better accuracy is obtained in the evaluation of displacement norm, energy norm and the stress intensity factor