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Scalable algorithms for physics-informed neural and graph networks
- Khemraj Shukla, Mengjia Xu, Nathaniel Trask, George E. Karniadakis
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- Journal:
- Data-Centric Engineering / Volume 3 / 2022
- Published online by Cambridge University Press:
- 29 June 2022, e24
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Physics-informed machine learning (PIML) has emerged as a promising new approach for simulating complex physical and biological systems that are governed by complex multiscale processes for which some data are also available. In some instances, the objective is to discover part of the hidden physics from the available data, and PIML has been shown to be particularly effective for such problems for which conventional methods may fail. Unlike commercial machine learning where training of deep neural networks requires big data, in PIML big data are not available. Instead, we can train such networks from additional information obtained by employing the physical laws and evaluating them at random points in the space–time domain. Such PIML integrates multimodality and multifidelity data with mathematical models, and implements them using neural networks or graph networks. Here, we review some of the prevailing trends in embedding physics into machine learning, using physics-informed neural networks (PINNs) based primarily on feed-forward neural networks and automatic differentiation. For more complex systems or systems of systems and unstructured data, graph neural networks (GNNs) present some distinct advantages, and here we review how physics-informed learning can be accomplished with GNNs based on graph exterior calculus to construct differential operators; we refer to these architectures as physics-informed graph networks (PIGNs). We present representative examples for both forward and inverse problems and discuss what advances are needed to scale up PINNs, PIGNs and more broadly GNNs for large-scale engineering problems.
Vortex-induced vibrations of a long flexible cylinder in shear flow
- REMI BOURGUET, GEORGE E. KARNIADAKIS, MICHAEL S. TRIANTAFYLLOU
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- Journal:
- Journal of Fluid Mechanics / Volume 677 / 25 June 2011
- Published online by Cambridge University Press:
- 27 April 2011, pp. 342-382
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We investigate the in-line and cross-flow vortex-induced vibrations of a long cylindrical tensioned beam, with length to diameter ratio L/D = 200, placed within a linearly sheared oncoming flow, using three-dimensional direct numerical simulation. The study is conducted at three Reynolds numbers, from 110 to 1100 based on maximum velocity, so as to include the transition to turbulence in the wake. The selected tension and bending stiffness lead to high-wavenumber vibrations, similar to those encountered in long ocean structures. The resulting vortex-induced vibrations consist of a mixture of standing and travelling wave patterns in both the in-line and cross-flow directions; the travelling wave component is preferentially oriented from high to low velocity regions. The in-line and cross-flow vibrations have a frequency ratio approximately equal to 2. Lock-in, the phenomenon of self-excited vibrations accompanied by synchronization between the vortex shedding and cross-flow vibration frequencies, occurs in the high-velocity region, extending across 30% or more of the beam length. The occurrence of lock-in disrupts the spanwise regularity of the cellular patterns observed in the wake of stationary cylinders in shear flow. The wake exhibits an oblique vortex shedding pattern, inclined in the direction of the travelling wave component of the cylinder vibrations. Vortex splittings occur between spanwise cells of constant vortex shedding frequency. The flow excites the cylinder under the lock-in condition with a preferential in-line versus cross-flow motion phase difference corresponding to counter-clockwise, figure-eight orbits; but it damps cylinder vibrations in the non-lock-in region. Both mono-frequency and multi-frequency responses may be excited. In the case of multi-frequency response and within the lock-in region, the wake can lock in to different frequencies at various spanwise locations; however, lock-in is a locally mono-frequency event, and hence the flow supplies energy to the structure mainly at the local lock-in frequency.
Turbulent drag reduction by constant near-wall forcing
- JIN XU, SUCHUAN DONG, MARTIN R. MAXEY, GEORGE E. KARNIADAKIS
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- Journal:
- Journal of Fluid Mechanics / Volume 582 / 10 July 2007
- Published online by Cambridge University Press:
- 14 June 2007, pp. 79-101
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Computational experiments based on the direct numerical simulation of turbulent channel flow reveal that the skin friction can be reduced as much as 70% by the action of a localized steady force acting against the flow close to the wall. In addition, the excessive shear stresses observed during the laminar-to-turbulence transition can be substantially reduced. For a sustained reduction in the skin friction, the control force has to act within a distance of 20 wall units (scaling with the location of the maximum Reynolds stress gradient); otherwise a transient drag reduction is observed or even an increase in drag. The forcing leads to the formation of a shear layer close to the wall that reduces the skin friction and limits the development of the Reynolds shear stresses. As the amplitude of the forcing is increased, the shear layer breaks down and generates its own turbulence, setting an upper limit to the level of drag reduction. This transition of the shear layer is correlated with a Reynolds number based on the forcing amplitude and length scale.
Three-dimensionality effects in flow around two tandem cylinders
- GEORGIOS V. PAPAIOANNOU, DICK K. P. YUE, MICHAEL S. TRIANTAFYLLOU, GEORGE E. KARNIADAKIS
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- Journal:
- Journal of Fluid Mechanics / Volume 558 / 10 July 2006
- Published online by Cambridge University Press:
- 04 July 2006, pp. 387-413
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The flow around two stationary cylinders in tandem arrangement at the laminar and early turbulent regime, ($\hbox{\it Re}\,{=}\,10^2$–$10^3$), is studied using two- and three-dimensional direct numerical simulations. A range of spacings between the cylinders from 1.1 to 5.0 diameters is considered with emphasis on identifying the effects of three-dimensionality and cylinder spacing as well as their coupling. To achieve this, we compare the two-dimensional with corresponding three-dimensional results as well as the tandem cylinder system results with those of a single cylinder. The critical spacing for vortex formation and shedding in the gap region depends on the Reynolds number. This dependence is associated with the formation length and base pressure suction variations of a single cylinder with Reynolds number. This association is useful in explaining some of the discrepancies between the two-dimensional and three-dimensional results. A major effect of three-dimensionality is in the exact value of the critical spacing, resulting in deviations from the two-dimensional predictions for the vorticity fields, the forces on the downstream cylinder, and the shedding frequency of the tandem system. Two-dimensional simulations under-predict the critical spacing, leading to erroneous results for the forces and shedding frequencies over a range of spacings where the flow is qualitatively different. To quantify the three-dimensional effects we first employ enstrophy, decomposed into a primary and a secondary component. The primary component involves the vorticity parallel to the cylinder axis, while the secondary component incorporates the streamwise and transverse components of the vorticity vector. Comparison with the single cylinder case reveals that the presence of the downstream cylinder at spacings lower than the critical value has a stabilizing effect on both the primary and secondary enstrophy. Systematic quantification of three-dimensionalities involves finding measures for the intensity of the spanwise fluctuations of the forces. This also verifies the stabilization scenario, suggesting that when the second cylinder is placed at a distance smaller than the critical one, three-dimensional effects are suppressed compared to the single-cylinder case. However, when the spacing exceeds the critical value, the upstream cylinder tends to behave like a single cylinder, but three-dimensionality in the flow generally increases.
Minimum-dissipation transport enhancement by flow destabilization: Reynolds’ analogy revisited
- George E. Karniadakis, Bora B. Mikic, Anthony T. Patera
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- Journal:
- Journal of Fluid Mechanics / Volume 192 / July 1988
- Published online by Cambridge University Press:
- 21 April 2006, pp. 365-391
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A classical transport enhancement problem is concerned with increasing the heat transfer in a system while minimizing penalties associated with shear stress, pressure drop, and viscous dissipation. It is shown by Reynolds' analogy that viscous dissipation in a wide class of flows scales linearly with the Nusselt number and quadratically with the Reynolds number. It thus follows that transport enhancement optimization is equivalent to a problem in hydrodynamic stability theory; a more unstable flow will achieve the same Nusselt number at a lower Reynolds number, and therefore at a fraction of the dissipative cost. This transport-stability theory is illustrated in a numerical study of supercritical (unsteady) two-dimensional flow in an eddy-promoter channel comprising a plane channel with an infinite periodic array of cylindrical obstructions.
It is shown that the addition of small cylinders to a plane channel results in stability modes that are little changed in form or frequency from plane-channel Tollmien-Schlichting waves. However, eddy-promoter flows are dramatically less stable than their plane-channel counterparts owing to cylinder-induced shear-layer instability (with critical Reynolds numbers on the order of hundreds rather than thousands), and thus these flows yield heat transfer rates commensurate with those of a plane-channel turbulent flow but at much lower Reynolds number. Small-cylinder supercritical eddy-promoter flows are shown to roughly preserve the convective-diffusive Reynolds analogy, and it thus follows from the transport-stability theory that eddy-promoter flows achieve the same heat transfer rates as plane-channel turbulent flows while incurring significantly less dissipation.