Non-intrusive nonlinear model reduction via machine learning approximations to low-dimensional operators

Although projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters; this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to $$10^3{\times }$$ 10 3 × in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs. The domain of applications include both parabolic and hyperbolic PDEs, regardless of the dimension of full-order models (FOMs).

Computational method for solving weakly singular Fredholm integral equations of the second kind using an advanced barycentric Lagrange interpolation formula

In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular Fredholm integral equations of the second kind. The kernel is interpolated twice concerning both variables and then is transformed into the product of five matrices; two of them are monomial basis matrices. To isolate the singularity of the kernel, we developed two techniques based on a good choice of different two sets of nodes to be distributed over the integration domain. Each set is specific to one of the kernel arguments so that the kernel values never become zero or imaginary. The significant advantage of thetwo presented techniques is the ability to gain access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the mean of the interpolant solution and the maximum error norm estimation are studied. The interpolate solutions of the illustrated four examples are found strongly converging uniformly to the exact solutions.

A separated representation involving multiple time scales within the Proper Generalized Decomposition framework

Solutions of partial differential equations can exhibit multiple time scales. Standard discretization techniques are constrained to capture the finest scale to accurately predict the response of the system. In this paper, we provide an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors. The proposed methodology is based on a time-separated representation within the standard Proper Generalized Decomposition, where the time coordinate is transformed into a multi-dimensional time through new separated coordinates, each representing one scale, while continuity is ensured in the scale coupling. For instance, when considering two different time scales, the governing Partial Differential Equation is commuted into a nonlinear system that iterates between the so-called microtime and macrotime, so that the time coordinate can be viewed as a 2D time. The macroscale effects are taken into account by means of a finite element-based macro-discretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the time domain. The resulting separated representation allows us a very fine time discretization without impacting the computational efficiency. The proposed formulation is explored and numerically verified on thermal and elastodynamic problems.

Evaluation of POD based surrogate models of fields resulting from nonlinear FEM simulations

Surrogate modelling is a powerful tool to replace computationally expensive nonlinear numerical simulations, with fast representations thereof, for inverse analysis, model-based control or optimization. For some problems, it is required that the surrogate model describes a complete output field. To construct such surrogate models, proper orthogonal decomposition (POD) can be used to reduce the dimensionality of the output data. The accuracy of the surrogate models strongly depends on the (pre)processing actions that are used to prepare the data for the dimensionality reduction. In this work, POD-based surrogate models with Radial Basis Function interpolation are used to model high-dimensional FE data fields. The effect of (pre)processing methods on the accuracy of the result field is systematically investigated. Different existing methods for surrogate model construction are compared with a novel method. Special attention is given to data fields consisting of several physical meanings, e.g. displacement, strain and stress. A distinction is made between the errors due to truncation and due to interpolation of the data. It is found that scaling the data per physical part substantially increases the accuracy of the surrogate model.

A simple yet consistent constitutive law and mortar-based layer coupling schemes for thermomechanical macroscale simulations of metal additive manufacturing processes

This article proposes a coupled thermomechanical finite element model tailored to the macroscale simulation of metal additive manufacturing processes such as selective laser melting. A first focus lies on the derivation of a consistent constitutive law on basis of a Voigt-type spatial homogenization procedure across the relevant phases, powder, melt and solid. The proposed constitutive law accounts for the irreversibility of phase change and consistently represents thermally induced residual stresses. In particular, the incorporation of a reference strain term, formulated in rate form, allows to consistently enforce a stress-free configuration for newly solidifying material at melt temperature. Application to elementary test cases demonstrates the validity of the proposed constitutive law and allows for a comparison with analytical and reference solutions. Moreover, these elementary solidification scenarios give detailed insights and foster understanding of basic mechanisms of residual stress generation in melting and solidification problems with localized, moving heat sources. As a second methodological aspect, dual mortar mesh tying strategies are proposed for the coupling of successively applied powder layers. This approach allows for very flexible mesh generation for complex geometries. As compared to collocation-type coupling schemes, e.g., based on hanging nodes, these mortar methods enforce the coupling conditions between non-matching meshes in an $$L^2$$ L 2 -optimal manner. The combination of the proposed constitutive law and mortar mesh tying approach is validated on realistic three-dimensional examples, representing a first step towards part-scale predictions.

Enhanced parametric shape descriptions in PGD-based space separated representations

Space separation within the Proper Generalized Decomposition—PGD—rationale allows solving high dimensional problems as a sequence of lower dimensional ones. In our former works, different geometrical transformations were proposed for addressing complex shapes and spatially non-separable domains. Efficient implementation of separated representations needs expressing the domain as a product of characteristic functions involving the different space coordinates. In the case of complex shapes, more sophisticated geometrical transformations are needed to map the complex physical domain into a regular one where computations are performed. This paper aims at proposing a very efficient route for accomplishing such space separation. A NURBS-based geometry representation, usual in computer aided design—CAD—, is retained and combined with a fully separated representation for allying efficiency (ensured by the fully separated representations) and generality (by addressing complex geometries). Some numerical examples are considered to prove the potential of the proposed methodology.

Industrial Digital Twins based on the non-linear LATIN-PGD

Digital Twins, which tend to intervene over the entire life cycle of products from early design phase to predictive maintenance through optimization processes, are increasingly emerging as an essential component in the future of industries. To reduce the computational time reduced-order modeling (ROM) methods can be useful. However, the spread of ROM methods at an industrial level is currently hampered by the difficulty of introducing them into commercial finite element software, due to the strong intrusiveness of the associated algorithms, preventing from getting robust and reliable tools all integrated in a certified product. This work tries to circumvent this issue by introducing a weakly-invasive reformulation of the LATIN-PGD method which is intended to be directly embedded into Simcenter Samcef$$^{\hbox {TM}}$$ TM finite element software. The originality of this approach lies in the remarkably general way of doing, allowing PGD method to deal with not only a particular application but with all facilities already included in such softwares—any non-linearities, any element types, any boundary conditions...—and thus providing a new high-performance all-inclusive non-linear solver.

Coupling reduced-order blood flow and cardiac models through energy-consistent strategies: modeling and discretization

In this work we provide a novel energy-consistent formulation for the classical 1D formulation of blood flow in an arterial segment. The resulting reformulation is shown to be suitable for the coupling with a lumped (0D) model of the heart that incorporates a reduced formulation of the actin-myosin interaction. The coupling being consistent with energy balances, we provide a complete heart-circulation model compatible with thermodynamics hence stable numerically and informative physiologically. These latter two properties are verified by numerical experiments.

A nonparametric probabilistic method to enhance PGD solutions with data-driven approach, application to the automated tape placement process

A nonparametric method assessing the error and variability margins in solutions depicted in a separated form using experimental results is illustrated in this work. The method assess the total variability of the solution including the modeling error and the truncation error when experimental results are available. The illustrated method is based on the use of the PGD separated form solutions, enriched by transforming a part of the PGD basis vectors into probabilistic one. The constructed probabilistic vectors are restricted to the physical solution’s Stiefel manifold. The result is a real-time parametric PGD solution enhanced with the solution variability and the confidence intervals.

Greedy maximin distance sampling based model order reduction of prestressed and parametrized abdominal aortic aneurysms

This work proposes a framework for projection-based model order reduction (MOR) of computational models aiming at a mechanical analysis of abdominal aortic aneurysms (AAAs). The underlying full-order model (FOM) is patient-specific, stationary and nonlinear. The quantities of interest are the von Mises stress and the von Mises strain field in the AAA wall, which result from loading the structure to the level of diastolic blood pressure at a fixed, imaged geometry (prestressing stage) and subsequent loading to the level of systolic blood pressure with associated deformation of the structure (deformation stage). Prestressing is performed with the modified updated Lagrangian formulation (MULF) approach. The proposed framework aims at a reduction of the computational cost in a many-query context resulting from model uncertainties in two material and one geometric parameter. We apply projection-based MOR to the MULF prestressing stage, which has not been presented to date. Additionally, we propose a reduced-order basis construction technique combining the concept of subspace angles and greedy maximin distance sampling. To further achieve computational speedup, the reduced-order model (ROM) is equipped with the energy-conserving mesh sampling and weighting hyper reduction method. Accuracy of the ROM is numerically tested in terms of the quantities of interest within given bounds of the parameter domain and performance of the proposed ROM in the many-query context is demonstrated by comparing ROM and FOM statistics built from Monte Carlo sampling for three different patient-specific AAAs.