Summary
In this work, we present an approach for the efficient treatment of parametrized geometries in the context of proper orthogonal decomposition (POD)‐Galerkin reduced order methods based on ...finite‐volume full order approximations. On the contrary to what is normally done in the framework of finite‐element reduced order methods, different geometries are not mapped to a common reference domain: the method relies on basis functions defined on an average deformed configuration and makes use of the discrete empirical interpolation method to handle together nonaffinity of the parametrization and nonlinearities. In the first numerical example, different mesh motion strategies, based on a Laplacian smoothing technique and on a radial basis function approach, are analyzed and compared on a heat transfer problem. Particular attention is devoted to the role of the nonorthogonal correction. In the second numerical example, the methodology is tested on a geometrically parametrized incompressible Navier‐Stokes problem. In this case, the reduced order model is constructed following the same segregated approach used at the full order level.
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SBCE, SBMB, UL, UM, UPUK
Obtaining an accurate mechanical model of a soft deformable robot compatible with the computation time imposed by robotic applications is often considered an unattainable goal. This paper should ...invert this idea. The proposed methodology offers the possibility to dramatically reduce the size and the online computation time of a finite element model (FEM) of a soft robot. After a set of expensive offline simulations based on the whole model, we apply snapshot-proper orthogonal decomposition to sharply reduce the number of state variables of the soft-robot model. To keep the computational efficiency, hyperreduction is used to perform the integration on a reduced domain. The method allows to tune the error during the two main steps of complexity reduction. The method handles external loads (contact, friction, gravity, etc.) with precision as long as they are tested during the offline simulations. The method is validated on two very different examples of FEMs of soft robots and on one real soft robot. It enables acceleration factors of more than 100, while saving accuracy, in particular compared to coarsely meshed FEMs and provides a generic way to control soft robots.
As the size of offshore wind turbines increases, a realistic representation of the spatiotemporal distribution of the incident wind field becomes crucial for modeling the dynamic response of the ...turbine. The International Electrotechnical Commission (IEC) standard for wind turbine design recommends two turbulence models for simulations of the incident wind field, the Mann spectral tensor model, and the Kaimal spectral and exponential coherence model. In particular, for floating wind turbines, these standard models are challenged by more sophisticated ones. The characteristics of the wind field depend on the stability conditions of the atmosphere, which neither of the standard turbulence models account for. The spatial and temporal distribution of the turbulence, represented by coherence, is not modeled consistently by the two standard models. In this study, the Mann spectral tensor model and the Kaimal spectral and exponential coherence model are compared with wind fields constructed from offshore measurements and obtained from large‐eddy simulations. Cross sections and durations relevant for offshore wind turbine design are considered. Coherent structures from the different simulators are studied across various stability conditions and wind speeds through coherence and proper orthogonal decomposition mode plots. As expected, the standard models represent neutral stratification better than they do stable and unstable. Depending upon the method used for generating the wind field, significant differences in the spatial and temporal distribution of coherence are found. Consequently, the computed structural design loads on a wind turbine are expected to vary significantly depending upon the employed turbulence model. The knowledge gained in this study will be used in future studies to quantify the effect of various turbulence models on the dynamic response of large offshore wind turbines.
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FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SBCE, SBMB, UL, UM, UPUK
This study emphasizes the flow mechanism of negative peak pressures on a square-section cylinder at Reynolds number of 22,000, particularly of those immediately upstream of the trailing edges ...(trailing-edge peak pressure). The minimum trailing-edge pressure coefficients are observed as low as -5 and have a close correlation in a streamwise length of approximately 0.1D (D is the width of the cylinder). The occurrence of small trailing-edge vortex, which has highly concentrated vorticity, is responsible for the trailing-edge peak pressures. The trailing-edge peak pressure occurs immediately before the shear layer moves far from the cylinder and recovers to the symmetric state. Standard proper orthogonal decomposition (POD) is applied to discover the inherent spatial modes with sharp peak pressures upstream of the trailing edges. Instead of the 1st mode (corresponding to vortex shedding), the higher-order modes contribute significantly to the trailing-edge peak pressures, whose mode coefficients have harmonics of vortex-shedding frequency. More than nine POD modes are necessary to reconstruct the trailing-edge peak pressures. As a variant of POD proposed recently, conditional space–time POD is valid in studying the average evolution of wind peak pressures. It further provides the relative phase and frequency ratio between the trailing-edge peak pressure and vortex shedding.
•Negative peak pressure on a 2D square-section cylinder at Re of 22,000 is studied.•Extremely low pressure is observed immediately upstream of trailing edges.•Flow mechanism of trailing-edge pressure is clarified due to trailing-edge vortex.•Average space–time characteristics of peak pressure are captured by conditional POD.•Inherent spatial modes responsible for peak pressure are found by standard POD.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We propose a nonlinear reduced basis method for the efficient approximation of parametrized partial differential equations (PDEs), exploiting kernel proper orthogonal decomposition (KPOD) for the ...generation of a reduced order space and neural networks for the evaluation of the reduced order approximation. In particular, we use KPOD in place of the more classical POD, on a set of high-fidelity solutions of the problem at hand to extract a reduced basis. This method provides a more accurate approximation of the snapshots' set featuring a lower dimension, while maintaining the same efficiency as POD. A neural network (NN) is then used to find the coefficients of the reduced basis by following a supervised learning paradigm and shown to be effective in learning the map between the time/parameter values and the projection of the high-fidelity snapshots onto the reduced space. In this NN, both the number of hidden layers and the number of neurons vary according to the intrinsic dimension of the differential problem at hand and the size of the reduced space. This adaptively built NN attains good performances in both the learning and the testing phases. Our approach is then tested on two benchmark problems, a one-dimensional wave equation and a two-dimensional nonlinear lid-driven cavity problem. We finally compare the proposed KPOD-NN technique with a POD-NN strategy, showing that KPOD allows a reduction of the number of modes that must be retained to reach a given accuracy in the reduced basis approximation. For this reason, the NN built to find the coefficients of the KPOD expansion is smaller, easier and less computationally demanding to train than the one used in the POD-NN strategy.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Because of its nonlinearity and path-dependency, analysis of the elasto-plastic behavior of the finite element (FE) model is computationally expensive. By directly learning sequential data, modeling ...plasticity via deep learning has shown successful performance in immediately predicting the path-dependent response. However, large-scale elasto-plastic FE models still have challenges in that they require numerous degrees of freedom and are accompanied by high-dimensional data. This study proposes a practical framework for the surrogate modeling of a large-scale elasto-plastic FE model by combining long short-term memory (LSTM) neural networks with proper orthogonal decomposition (POD). First, displacement, plastic strain magnitude, and von Mises stress are generated using commercial FE analysis software, and then, the high-dimensional data are reduced to low-dimensional POD coefficient data before being used for training. With the drastically reduced data, a neural network architecture can be introduced in the form of individual and ensemble structures to achieve accurate and robust prediction. As the proposed POD-LSTM surrogate model operates on the structural level, POD-LSTM surrogate models are constructed and implemented for each of the three large-scale elasto-plastic FE models. In all three examples, the proposed POD-LSTM surrogate models were found to be efficient and accurate for predicting elasto-plastic responses.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
•Two model-reduction methods that project dynamical systems on nonlinear manifolds.•Analysis including conditions under which the two methods are equivalent.•A novel convolutional autoencoder ...architecture to construct the nonlinear manifold.•Numerical experiments demonstrating the method outperforms linear subspaces.
Nearly all model-reduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the reduced-basis method, and (balanced) proper orthogonal decomposition (POD). Unfortunately, restricting the state to evolve in a linear subspace imposes a fundamental limitation to the accuracy of the resulting reduced-order model (ROM). In particular, linear-subspace ROMs can be expected to produce low-dimensional models with high accuracy only if the problem admits a fast decaying Kolmogorov n-width (e.g., diffusion-dominated problems). Unfortunately, many problems of interest exhibit a slowly decaying Kolmogorov n-width (e.g., advection-dominated problems). To address this, we propose a novel framework for projecting dynamical systems onto nonlinear manifolds using minimum-residual formulations at the time-continuous and time-discrete levels; the former leads to manifold Galerkin projection, while the latter leads to manifold least-squares Petrov–Galerkin (LSPG) projection. We perform analyses that provide insight into the relationship between these proposed approaches and classical linear-subspace reduced-order models; we also derive a posteriori discrete-time error bounds for the proposed approaches. In addition, we propose a computationally practical approach for computing the nonlinear manifold, which is based on convolutional autoencoders from deep learning. Finally, we demonstrate the ability of the method to significantly outperform even the optimal linear-subspace ROM on benchmark advection-dominated problems, thereby demonstrating the method's ability to overcome the intrinsic n-width limitations of linear subspaces.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
•We study the flow characteristics in a tightly packed rod bundle with wire spacers.•SPIV measurements were taken on a cross-flow plane, measured area covered subchannels.•Vorticity vectors show flow ...patterns: swirls, wakes, and recirculation in subchannels.•Quasi-instantaneous 3D vortical structures were identified by vorticity iso-surfaces.•Coherent structures in subchannels were revealed by POD analysis to velocity fields.
In this study, we experimentally investigated the flow field characteristics in the complex geometry of a tightly packed 61-rod bundle with wire spacers by employing matched-index-of-refraction (MIR) and stereoscopic particle image velocimetry (SPIV) techniques. SPIV measurements were performed on a cross-flow plane at Reynolds numbers of 500, 2500, and 6300, and the measurement area covered the interior and edge sub-channels that were located near the enclosure wall. The SPIV results showed that peaks of the velocity magnitude occurred in the edge sub-channel, indicating the wall effects on the flow symmetry in the cross-flow plane. Mean velocities at the central areas of the edge and interior sub-channels showed flat profiles for Re=500 and local peaks around 0.15<y/Drod<0.3, due to the wire’s presence and near-wall location of the edge sub-channel. The rms fluctuating profiles velocities showed that local maximal peaks linearly shifted from y/Drod=-0.44 to −0.36 and −0.28 as Re increased. Taylor’s hypothesis was applied to the SPIV velocity vector fields for reconstructing quasi-instantaneous three-dimensional (3D) vortical structures in the wire-wrapped rod bundle. The visualization of 3D vorticity iso-surfaces showed that vortical structures were generated from shear layers of neighboring rods and traveled in the streamwise direction. Finally, proper orthogonal decomposition (POD) analysis was applied to SPIV velocity snapshots taken at the edge and interior sub-channels to extract the most statistically dominant flow structures. The POD velocity decomposition revealed coherent structures with sizes and shapes comparable to instantaneous vortical structures identified by iso-surfaces. Energy fractions of the first POD modes were found to reduce from 46% and 31% to 26% and 18% when Reynolds numbers increased from Re=500 to 6300, respectively. On the other hand, kinetic energy levels of low-order POD modes (modes 2, 3, 4, etc.) increased when the Reynolds number increased.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK, ZRSKP
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