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Stabilized finite element formulations for flow problems.

機(jī)譯：針對(duì)流動(dòng)問(wèn)題的穩(wěn)定有限元公式。

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This work focuses on developing novel stabilized finite element formulations for flow problems suitable for monolithic approaches to fluid-structure interaction. In addition to the challenges produced by the poor performance of the classical Galerkin method for mixed finite elements, monolithic methods for nonlinear problems require consistent linearization. One of the main contributions of the work is a consistent linearization of a variational multiscale formulation for incompressible Navier-Stokes. We compare the convergence of this formulation with a fixed-point iteration technique. We show for a number of example problems that the Newton-Raphson technique converges quadratically. For transient problems, we show that consistent Newton-Raphson approach does not converge in the vicinity of bifurcations such as the onset of unsteady vortex shedding. We also present a Schur's complement implementation of the consistent method that may be considered an extension of static condensation to nonlinear problems. The Newton-Schur technique greatly increases parallel performance, which helps offset the computational cost produced by the decomposition of the multiscale framework. For a number of three-dimensional problems (at Reynolds number up to 1000) we show that the Newton-Schur approach is scalable for reasonable problem sizes. To investigate the central challenges to numerically modeling the Navier-Stokes equations, we begin with a study of the Stokes and advection-diffusion equations. From our study of the Stokes equations we learn that stabilized methods based on enriching the velocity trial function space with bubble functions are only stable for triangular elements. We also present a stabilized formulation for the advection-diffusion equation using the generalized finite element framework. This formulation eliminates oscillations in the neighborhood of sharp gradients and requires fewer degrees of freedom than variational multiscale methods.
機(jī)譯：這項(xiàng)工作的重點(diǎn)是針對(duì)流動(dòng)問(wèn)題開(kāi)發(fā)適用于整體方法進(jìn)行流固耦合的新型穩(wěn)定有限元公式。除了經(jīng)典的Galerkin方法對(duì)混合有限元的性能不佳帶來(lái)的挑戰(zhàn)之外，用于非線性問(wèn)題的整體方法還要求一致的線性化。這項(xiàng)工作的主要貢獻(xiàn)之一是對(duì)不可壓縮的Navier-Stokes的變分多尺度公式進(jìn)行了一致的線性化。我們用定點(diǎn)迭代技術(shù)比較了該公式的收斂性。我們以牛頓-拉夫森技術(shù)為平方收斂的許多示例問(wèn)題進(jìn)行了展示。對(duì)于瞬態(tài)問(wèn)題，我們表明一致的Newton-Raphson方法不會(huì)在分叉附近收斂，例如不穩(wěn)定渦旋脫落的發(fā)生。我們還提出了一致方法的Schur's補(bǔ)充實(shí)現(xiàn)，可以將其視為靜態(tài)凝聚對(duì)非線性問(wèn)題的擴(kuò)展。牛頓-舒爾技術(shù)極大地提高了并行性能，這有助于抵消多尺度框架分解產(chǎn)生的計(jì)算成本。對(duì)于許多三維問(wèn)題（雷諾數(shù)最大為1000），我們證明了牛頓舒爾方法對(duì)于合理的問(wèn)題大小是可擴(kuò)展的。為了研究對(duì)Navier-Stokes方程進(jìn)行數(shù)值建模的主要挑戰(zhàn)，我們從研究Stokes和對(duì)流擴(kuò)散方程開(kāi)始。從我們對(duì)Stokes方程的研究中，我們發(fā)現(xiàn)基于氣泡函數(shù)豐富速度試驗(yàn)函數(shù)空間的穩(wěn)定化方法僅對(duì)三角形元素穩(wěn)定。我們還提出了使用廣義有限元框架的對(duì)流擴(kuò)散方程的穩(wěn)定公式。與變分多尺度方法相比，該公式消除了陡峭梯度附近的振蕩，并且所需的自由度更少。

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