Adaptive Model Order Reduction Research Articles

Data-driven model order reduction for granular media

We investigate the use of reduced-order modelling to run discrete element simulations at higher speeds. Taking a data-driven approach, we run many offline simulations in advance and train a model to predict the velocity field from the mass distribution and system control signals. Rapid model inference of particle velocities replaces the intense process of computing contact forces and velocity updates. In coupled DEM and multibody system simulation, the predictor model can be trained to output the interfacial reaction forces as well. An adaptive model order reduction technique is investigated, decomposing the media in domains of solid, liquid, and gaseous state. The model reduction is applied to solid and liquid domains where the particle motion is strongly correlated with the mean flow, while resolved DEM is used for gaseous domains. Using a ridge regression predictor, the performance is tested on simulations of a pile discharge and bulldozing. The measured accuracy is about 90% and 65%, respectively, and the speed-up range between 10 and 60.

Model order reduction for small strain elasto‐plasticity based on an adaptive strategy

AbstractIn this contribution, an online adaptive reduced order model is presented for small‐strain elasto‐plasticity. Using the GALERKIN method, the problem is projected to a lower subspace giving a reduced problem based on the displacement field snapshots. The adaptivity lies in the update of the projection operator, which is carried out in different parts of the simulation if certain criteria related to the convergence behavior are not fulfilled. With a relative POD analysis called Proper Snapshot Selection (PSS), the new basis is chosen and the simulation is proceeded.

An adaptive model order reduction method for boundary element-based multi-frequency acoustic wave problems

The classical boundary element method (BEM) is widely used to obtain detailed information on the acoustic performance of large-scale dynamical systems due to the nature of semi-analytical characteristic. Its use, however, results in asymmetric and dense system matrix, which makes original full-order model evaluation very time consuming and memory demanding. Moreover, the frequency sweep analysis which is indispensable for the assessment of noise emission levels and the design of high-quality products requires the repetitive assembly and solution of system of equations, which further increases the computational complexity. In order to alleviate these problems, an adaptive structure-preserving model order reduction method is presented, which is based on an offline–online solution framework. In the offline phase, we first factor out the frequency term as a scalar function from the BE integral kernels of the Burton–Miller formulation, followed by integration to set up the system matrices. A global frequency-independent orthonormal basis is then constructed via the second-order Arnoldi (SOAR) method to span a projection subspace, onto which the frequency-decoupled system matrices are projected column by column to solve the memory problem arising from the frequency-related decomposition. In addition, the number of iterations required for convergence can be automatically determined by exploiting the condition number of an upper Hessenberg matrix in SOAR. In the online stage, a reliable reduced-order model can be quickly recovered by the sum of those offline stored reduced matrices multiplied by frequency-dependent coefficients, which is favored in many-query applications. Two academic benchmarks and a more realistic problem are investigated in order to demonstrate the potentials of the proposed approach.

A discontinuous and adaptive reduced order model for the angular discretization of the Boltzmann transport equation

SummaryThis article presents a new adaptive reduced order model for resolving the angular direction of the Boltzmann transport equation, based on proper orthogonal decomposition (POD) and the method of snapshots. It builds upon previous methods of applying POD to the angular dimension, with modifications to increase accuracy and solver stability. Previous methods used continuous global functions spanning the whole sphere. The new approach, discontinuous POD (DPOD), partitions the surface of the sphere into angular regions, each with an independent set of POD basis functions. Combined, these can approximate flux distributions which span the sphere using optimized basis functions for each angular region. In addition, a novel implementation of adaptive angular resolution known as adaptive discontinuous POD (ADPOD) is presented, which allows the number of DPOD basis functions to vary by angular octant and spatial element. DPOD and ADPOD are applied to two problems in order to demonstrate their benefits compared with POD. Both are shown to reduce the number of solver iterations required to find a solution and decrease the error in the angular flux.

On an improved adaptive reduced-order model for the computation of steady-state vibrations in large-scale non-conservative systems with friction joints

Joints are commonly used in many large-scale engineering systems to ease assembly, and ensure structural integrity and effective load transmission. Most joints are designed around friction interfaces, which can transmit large static forces, but tend to introduce stick-slip transition during vibrations, leading to a nonlinear dynamic system. Tools for the complex numerical prediction of such nonlinear systems are available today, but their use for large-scale applications is regularly prevented by high computational cost. To address this issue, a novel adaptive reduced-order model (ROM) has recently been developed, significantly decreasing the computational time for such high fidelity simulations. Although highly effective, significant improvements to the proposed approach is presented and demonstrated in this paper, further increasing the efficiency of the ROM. An energy-based error estimator was developed and integrated into the nonlinear spectral analysis, leading to a significantly higher computational speed by removing insignificant static modes from the stuck contact nodes in the original reduced basis, and improving the computational accuracy by eliminating numerical noise. The effectiveness of the new approach was shown on an industrial-scale fan blades system with a dovetail joints, showing that the improved adaptive method can be 2–3 times more computationally efficient than the original adaptive method especially at high excitation levels but also effectively improve the accuracy of the original method.

Acoustic simulation of cavities with porous materials using an adaptive model order reduction technique

Porous materials are often used in the context of passive noise control applications due to their low cost and good sound absorption properties. As the most commonly used predictive tool, the finite element (FE) method is often applied in the design phase to obtain detailed information on the performance of complex vibro-acoustic systems with porous damping treatments. In this case, however, its use inevitably leads to very large, complex-valued and frequency-dependent numerical models, which makes original full-order model evaluation intractable due to time and memory limitations. In this paper, an adaptive model order reduction strategy is presented to reduce the number of degrees of freedom involved in the associated problems such that the required computational cost can be largely alleviated while a desired high accuracy can be achieved in a controllable way. This technique consists of making use of Taylor expansion to the multiple frequency-dependent scalar functions coming from the complex material behavior, and then performing the structure-preserving second-order Arnoldi algorithm based on a robust error indicator to solve the underlying FE model in the frequency domain. A further improvement in terms of the approximation accuracy of the constructed compact reduced-order model is also proposed to incorporate all available moment-matching information. Two widely used model types for porous materials are investigated, i.e. the equivalent fluid description and the Biot theory. Mono-physical poroelastic-poroelastic coupling and multi-physical acoustic-porous coupling are considered in order to demonstrate the simplicity, versatility and efficiency of the approach.

Adaptive reduced order modeling for nonlinear dynamical systems through a new a posteriori error estimator: Application to uncertainty quantification

SummaryMultiquery problems such as uncertainty quantification (UQ), optimization of a dynamical system require solving a differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. This issue can be addressed by using a cheaper reduced order model (ROM) instead. However, the ROM entails error in the solution due to approximation in a lower dimensional subspace. Moreover, the ROM lacks robustness over a wide range of parameter values. To address these issues, first, an upper bound on the norm of the state transition matrix is derived. This bound, along with the residual in the governing equation, are then used to develop an error estimator for general nonlinear dynamical systems. Furthermore, this error estimator is used in conjunction with the modified greedy search algorithm proposed by Hossain and Ghosh (Int J Numer Methods Eng, 2018;116(12‐13): 741‐758) to adaptively construct a robust proper orthogonal decomposition‐based ROM. This adaptive ROM is subsequently deployed for UQ by invoking it in a statistical simulation. Two numerical studies: (i) viscous Burgers' equation and (ii) beam on nonlinear Winkler foundation, showed an improved accuracy of the error estimator compared to the current literature. A significant computational speed‐up in UQ is achieved.

A nonintrusive adaptive reduced order modeling approach for a molten salt reactor system

We use a novel nonintrusive adaptive Reduced Order Modeling method to build a reduced model for a molten salt reactor system. Our approach is based on Proper Orthogonal Decomposition combined with locally adaptive sparse grids. Our reduced model captures the effect of 27 model parameters on keff of the system and the spatial distribution of the neutron flux and salt temperature. The reduced model was tested on 1000 random points. The maximum error in multiplication factor was found to be less than 50 pcm and the maximum L2 error in the flux and temperature were less than 1%. Using 472 snapshots, the reduced model was able to simulate any point within the defined range faster than the high-fidelity model by a factor of 5×106. We then employ the reduced model for uncertainty and sensitivity analysis of the selected parameters on keff and the maximum temperature of the system.

Accelerating crack growth simulations through adaptive model order reduction

SummaryAccurate numerical modeling of fracture in solids is a challenging undertaking that often involves the use of computationally demanding modeling frameworks. Model order reduction techniques can be used to alleviate the computational effort associated with these models. However, the traditional offline‐online reduction approach is unsuitable for complex fracture phenomena due to their excessively large parameter spaces. In this work, we present a reduction framework for fracture simulations that leaves out the offline training phase and instead adaptively constructs reduced solutions spaces online. We apply the framework to the thick level set (TLS) method, a novel approach for modeling fracture able to model crack initiation, propagation, branching, and merging. The analysis starts with a fully‐solved load step, after which two consecutive reduction operations—the proper orthogonal decomposition and the empirical cubature method—are performed. Numerical features specific to the TLS method are used to define an adaptive domain decomposition scheme that allows for three levels of model reduction coexisting on the same finite element mesh. Special solutions are proposed that allow the framework to deal with enriched nodes and a dynamic number of integration points. We demonstrate and assess the performance of the framework with a number of numerical examples.

An Adaptive Model Order Reduction Method Based on the Damage Evolution for Nonlinear Seismic Analysis

Nonlinear seismic analysis, an approach to evaluate the seismic performance of a structure, is facing the challenge of computational efficiency for large‐scale and high‐fidelity simulation. This paper proposes an adaptive model order reduction (MOR) method based on the damage evolution among the overall structure to alleviate the computational burden. The damage state of each component during seismic loadings is distinguished as the initial‐elastic phase, the plastic‐damage phase, and the residual‐elastic phase. In order to exploit the potential of model order reduction based on the damage evolution, a duration spectrum analysis is utilized to evaluate the characteristics of the residual‐elastic phase for SDOF systems with bilinear hysteretic behaviour. Thus, an adaptive MOR method has been proposed to handle the nonlinear dynamic analysis of structures during different damage evolution phases. The overall structure is adaptively partitioned into linear substructures and nonlinear substructures on the basis of the time‐varying damage distribution. The model order of linear substructures is reduced using the initial stiffness‐based vibration modes, while nonlinear substructures that keep in the residual‐elastic phase are reduced using the tangent‐stiffness‐based vibration modes. The residual displacements of nonlinear substructures are treated as the initial deformation during the residual‐elastic phase. Compared with the traditional time step integration method, the proposed adaptive MOR method is able to increase the computational efficiency as yielding comparative results.

Adaptive Reduced-Order Model Construction for Conditional Value-at-Risk Estimation

This paper shows how to systematically and efficiently improve a reduced-order model (ROM) to obtain a better ROM-based estimate of the Conditional Value-at-Risk (CVaR) of a computationally expensi...

Explicit and efficient topology optimization of frequency-dependent damping patches using moving morphable components and reduced-order models

Viscoelastic materials have been widely applied as the lightweight structures are growingly used in many industrial sectors in order to improve the cost-efficiency of products. The present work is devoted to minimize the kinetic energy of thin-shell structures subjected to harmonic excitations within a certain frequency range by seeking the optimal layouts of frequency-dependent viscoelastic damping patches under a prescribed area constraint. An explicit topology optimization approach based on the moving morphable components (MMC) is applied, where the number of design variables is substantially reduced compared to the traditional density-based implicit framework. This is achieved by introducing a set of geometry parameters (i.e. design variables) to explicitly describe the boundary of structural components which are viscoelastic patches in this paper. Meanwhile, the unnecessary degrees of freedom (DOFs) related to void regions from the finite element (FE) model are removed at every step of numerical optimization thanks to the explicit boundary evolution. The adjoint method is adopted to derive the sensitivity of the objective function with respect to a design variable. Furthermore, an adaptive model order reduction (MOR) technique for the frequency-dependent system is provided to simplify the computational complexity of the dynamical equation and the adjoint equation as well. With the MMC to reduce the number of design variables in the topology formulation and the MOR to reduce the number of DOFs in the FE model, the optimization simulation can be largely sped up. Numerical examples are presented to demonstrate that the combination of the MMC- and MOR-technique is able to distribute constrained-layer damping patches reasonably and very efficiently.

An Error Indicator-Based Adaptive Reduced Order Model for Nonlinear Structural Mechanics—Application to High-Pressure Turbine Blades

The industrial application motivating this work is the fatigue computation of aircraft engines’ high-pressure turbine blades. The material model involves nonlinear elastoviscoplastic behavior laws, for which the parameters depend on the temperature. For this application, the temperature loading is not accurately known and can reach values relatively close to the creep temperature: important nonlinear effects occur and the solution strongly depends on the used thermal loading. We consider a nonlinear reduced order model able to compute, in the exploitation phase, the behavior of the blade for a new temperature field loading. The sensitivity of the solution to the temperature makes the classical unenriched proper orthogonal decomposition method fail. In this work, we propose a new error indicator, quantifying the error made by the reduced order model in computational complexity independent of the size of the high-fidelity reference model. In our framework, when the error indicator becomes larger than a given tolerance, the reduced order model is updated using one time step solution of the high-fidelity reference model. The approach is illustrated on a series of academic test cases and applied on a setting of industrial complexity involving five million degrees of freedom, where the whole procedure is computed in parallel with distributed memory.

A certified model reduction approach for robust parameter optimization with PDE constraints

We investigate an optimization problem governed by an elliptic partial differential equation with uncertain parameters. We introduce a robust optimization framework that accounts for uncertain model parameters. The resulting non-linear optimization problem has a bi-level structure due to the min-max formulation. To approximate the worst-case in the optimization problem we propose linear and quadratic approximations. However, this approach still turns out to be very expensive, therefore we propose an adaptive model order reduction technique which avoids long offline stages and provides a certified reduced order surrogate model for the parametrized PDE which is then utilized in the numerical optimization. Numerical results are presented to validate the presented approach.

A Fast Leakage-Aware Full-Chip Transient Thermal Estimation Method

Accurate and fast thermal estimation is important for the runtime thermal regulation of modern microprocessors due to excessive on-chip temperatures. However, due to the nonlinear relationship between the leakage power and temperature, full-chip thermal estimation methods suffer slow speed and scalability issue when the increasing static leakage power is considered. In this work, we propose a new fast leakage-aware full-chip thermal estimation method. Unlike traditional methods, which use iteration to handle the leakage-temperature nonlinearity dependency issue, the new method applies a dynamic linearization algorithm, which adaptively transforms the original nonlinear thermal model into a number of local linear thermal models. In order to further improve the thermal estimation efficiency, a specially-designed adaptive model order reduction method is integrated into the thermal estimation framework to generate local compact thermal models. Our numerical results show that the new method is able to accurately estimate full-chip transient temperature distribution by fully considering the nonlinear leakage-temperature dependency with fast speed. On different chips with core number ranging from 9 to 36, it achieved 85x to 589x speedup in average against traditional iteration based method, with average thermal estimation error to be around 0.2°C.

Adaptive parametric model order reduction technique for optimization of vibro-acoustic models: Application to hearing aid design

Finite Element (FE) models of complex structural-acoustic coupled systems can require a large number of degrees of freedom in order to capture their physical behaviour. This is the case in the hearing aid field, where acoustic-mechanical feedback paths are a key factor in the overall system performance and modelling them accurately requires a precise description of the strong interaction between the light-weight parts and the internal and surrounding air over a wide frequency range. Parametric optimization of the FE model can be used to reduce the vibroacoustic feedback in a device during the design phase; however, it requires solving the model iteratively for multiple frequencies at different parameter values, which becomes highly time consuming when the system is large. Parametric Model Order Reduction (pMOR) techniques aim at reducing the computational cost associated with each analysis by projecting the full system into a reduced space. A drawback of most of the existing techniques is that the vector basis of the reduced space is built at an offline phase where the full system must be solved for a large sample of parameter values, which can also become highly time consuming. In this work, we present an adaptive pMOR technique where the construction of the projection basis is embedded in the optimization process and requires fewer full system analyses, while the accuracy of the reduced system is monitored by a cheap error indicator. The performance of the proposed method is evaluated for a 4-parameter optimization of a frequency response for a hearing aid model, evaluated at 300 frequencies, where the objective function evaluations become more than one order of magnitude faster than for the full system.

Adaptive Multipoint Model Order Reduction Scheme for Large-Scale Inductive PEEC Circuits

The model order reduction techniques based on the multipoint projection Krylov methods have become the methods of choice to generate large macromodels of multiport RLC circuits. A well-known difficulty with such methods lies in the need for clever point selection to attain model compactness and accuracy. In this paper, we present an automatic methodology for optimizing sample point selection. This method is general, suitable for circuits obtained by the partial element equivalent circuit method coupled with an adaptive multilevel fast multipole method. Our algorithm has been validated on an industrial example to demonstrate the accuracy and robustness of the approach.

Simplex‐based adaptive parametric model order reduction for applications in optimization

AbstractA new methodology for optimization using parametric reduced order models is introduced. An adaptive scheme to place the expansion points in the reduction step is combined with a smart parametric space sampling in the optimization step to produce very reliable reduced order models. The methodology is illustrated in a demonstrative electromagnetic problem with appreciable gain in computational time. This technique is of general application and can be used to speed up optimization processes in many areas of engineering.

Computationally efficient simulation of unsteady aerodynamics using POD on the fly

Modern industrial aircraft design requires a large amount of sufficiently accurate aerodynamic and aeroelastic simulations. Current computational fluid dynamics (CFD) solvers with aeroelastic capabilities, such as the NASA URANS unstructured solver FUN3D, require very large computational resources. Since a very large amount of simulation is necessary, the CFD cost is just unaffordable in an industrial production environment and must be significantly reduced. Thus, a more inexpensive, yet sufficiently precise solver is strongly needed. An opportunity to approach this goal could follow some recent results (Terragni and Vega 2014 SIAM J. Appl. Dyn. Syst. 13 330–65; Rapun et al 2015 Int. J. Numer. Meth. Eng. 104 844–68) on an adaptive reduced order model that combines ‘on the fly’ a standard numerical solver (to compute some representative snapshots), proper orthogonal decomposition (POD) (to extract modes from the snapshots), Galerkin projection (onto the set of POD modes), and several additional ingredients such as projecting the equations using a limited amount of points and fairly generic mode libraries. When applied to the complex Ginzburg–Landau equation, the method produces acceleration factors (comparing with standard numerical solvers) of the order of 20 and 300 in one and two space dimensions, respectively. Unfortunately, the extension of the method to unsteady, compressible flows around deformable geometries requires new approaches to deal with deformable meshes, high-Reynolds numbers, and compressibility. A first step in this direction is presented considering the unsteady compressible, two-dimensional flow around an oscillating airfoil using a CFD solver in a rigidly moving mesh. POD on the Fly gives results whose accuracy is comparable to that of the CFD solver used to compute the snapshots.

Null space correction and adaptive model order reduction in multi-frequency Maxwell’s problem

A model order reduction method is developed for an operator with a non-empty null-space and applied to numerical solution of a forward multi-frequency eddy current problem using a rational interpolation of the transfer function in the complex plane. The equation is decomposed into the part in the null space of the operator, calculated exactly, and the part orthogonal to it which is approximated on a low-dimensional rational Krylov subspace. For the Maxwell's equations the null space is related to the null space of the curl. The proposed null space correction is related to divergence correction and uses the Helmholtz decomposition. In the case of the finite element discretization with the edge elements, it is accomplished by solving the Poisson equation on the nodal elements of the same grid. To construct the low-dimensional approximation we adaptively choose the interpolating frequencies, defining the rational Krylov subspace, to reduce the maximal approximation error. We prove that in the case of an adaptive choice of shifts, the matrix spanning the approximation subspace can never become rank deficient. The efficiency of the developed approach is demonstrated by applying it to the magnetotelluric problem, which is a geophysical electromagnetic remote sensing method used in mineral, geothermal, and groundwater exploration. Numerical tests show an excellent performance of the proposed methods characterized by a significant reduction of the computational time without a loss of accuracy. The null space correction regularizes the otherwise ill-posed interpolation problem.