Abstract

Model reduction in computational mechanics is generally addressed with linear dimensionality reduction methods such as Principal Components Analysis (PCA). Hypothesizing that in many applications of interest the essential dynamics evolve on a nonlinear manifold, we explore here reduced order modeling based on nonlinear dimen- sionality reduction methods. Such methods are gaining popularity in diverse fields of science and technology, such as machine perception or molecular simulation. We consider finite deformation elastodynamics as a model problem, and identify the manifold where the dynamics essentially take place –the slow manifold– by nonlinear dimensionality reduction methods applied to a database of snapshots. Contrary to linear dimensionality reduction, the smooth parametrization of the slow manifold needs special techniques, and we use local maximum entropy approximants. We then formulate the Lagrangian mechanics on these data-based generalized coordinates, and de- velop variational time-integrators. Our proof-of-concept example shows that a few nonlinear collective variables provide similar accuracy to tens of PCA modes, suggesting that the proposed method may be very attractive in control or optimization applications. Furthermore, the reduced number of variables brings insight into the me- chanics of the system under scrutiny. Our simulations also highlight the need of modeling the net e ¿ ect of the disregarded degrees of freedom on the reduced dynamics at long times.

Abstract

Tensor representations allow compact storage and efficient manipulation of multi-dimensional data. Based on these, tensor methods build low-rank subspaces for the solution of multi-dimensional and multi-parametric models. However, tensor methods cannot always be implemented efficiently, specially when dealing with non-linear models. In this paper, we discuss the importance of achieving a tensor representation of the model itself for the efficiency of tensor-based algorithms. We investigate the adequacy of interpolation rather than projection-based approaches as a means to enforce such tensor representation, and propose the use of cross approximations for models in moderate dimension. Finally, linearization of tensor problems is analyzed and several strategies for the tensor subspace construction are proposed. This is a post-peer-review, pre-copyedit version of an article published in Journal of scientific computing

Abstract

Beam-like components help to provide structural integrity in a wide range of applications. Initially made of alloys or natural materials such as wood, today’s technologies like pultrusion make possible the manufacturing of such components with composite materials providing good quality products with high performance to weight ratio. The anisotropic nature of composite materials, though, poses a challenging framework when numerically simulating them. This work presents the integration in standard Finite Element (FE) packages of a machine learning methodology that naturally captures the behaviour of composite materials: the multiscale method for periodic structures using domain decomposition and ECM-hyper reduction. The method provides a special type of finite element that can be assembled using FE libraries. However, the kinematics of this element may be described by more Degrees of Freedom (DoF) per node than most standard FE packages consider, hence implementing it is a non-trivial task. The strategy presented here tries to tackle the limitation by means of static condensation followed by a regression procedure. Periodic boundary conditions are employed for the extra DoFs while the regression consists in a polynomial on the logarithmic space. This paper focuses on the performance of the integrated beam element through the analysis of a pultruded omega profile whose anisotropy requires the use of complex models to accurately capture its mechanical behaviour. The agreement between the results obtained with the proposed model and those given by more complex formulations validates the methodology and enables its use in regular FE codes to characterize complex composite beam structures.

Abstract

Modal analysis of three-dimensional gravitational thin liquid sheet flows, interacting with unconfined gaseous environments located on both sides of the liquid phase, is performed in the present work. Numerical data of this relevant two-phase flow configuration are obtained through the single-phase formulation and the Volume-of-Fluid (VOF) technique implemented in the flow solver Basilisk. This class of flows exhibits a variety of spatial and temporal relevant structures, both in free and forced configurations, that are investigated through the Spectral Proper Orthogonal Decomposition (SPOD). By means of such methodology, we explore the effect of two main governing parameters on the flow dynamics, namely the liquid sheet aspect ratio, AR = W/H, where H and W are the sheet inlet thickness and width, and the Weber number, We = lU2H/(2), in which U is the inlet liquid velocity, lthe liquid density, and the surface tension coefficient. Finally, for the highest aspect ratio value considered (AR = 40), we investigate the forced dynamics of the system excited by a harmonic perturbation in transverse velocity component applied at the inlet section, comparing results with ones arising from a purely two-dimensional analysis of the flow. The obtained results highlight the low rank behavior exhibited by the flow, suggesting that reduced order modeling could be particularly appealing to reduce complexity and computational effort in numerical simulation of this class of flows.