|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| The '''projection method''' is an effective means of [[Numerical analysis|numerically]] solving time-dependent [[incompressible flow|incompressible fluid-flow]] problems. It was originally introduced by [[Alexandre Chorin]] in 1967 and independently by [[Roger Temam]]<ref>
| | Hi, everybody! My name is Theresa. <br>It is a little about myself: I live in Austria, my city of Pollach. <br>It's called often Eastern or cultural capital of SALZBURG. I've married 4 years ago.<br>I have 2 children - a son (Columbus) and the daughter (Noah). We all like Amateur radio.<br>xunjie 風の動きが最先端のニューヨーク、 |
| {{Citation
| | 同社は日本と西欧およびその他の国から一流のプロ子供服の生産設備を持っている以上の5台、 |
| | surname1 = Temam
| | 地元やその他の状況を支援するリーディングカンパニーによると、 [http://www.dressagetechnique.com/images/jp/top/jimmychoo/ ���ߩ`��奦 �ȩ`�ȥХå�] カナリ2012年秋と冬のシリーズは、 |
| | given1 = R.
| | すべてのスタッフの出店商売繁盛におめでとうと言いたいが活況を呈しています。 |
| | title = Une méthode d'approximation des solutions des équations Navier-Stokes,
| | 糸くずのブーツのすべての種類のための時間で組織になりましたUggのブーツは、 [http://www.equityfair.ch/gzd/jr/mall/shoe/newbalance/ �˥�`�Х�� ���˩`���` ���] 例外的に魅力的なカーブを作成するのに役立つことができることを感じさせるだけではありません。 |
| | journal = Bull. Soc. Math. France
| | 完全にマルチアングルディスプレイは英国のファッションオリエンタルスタイルに属します。 |
| | volume = 98
| | プラス私たちが作成するためのクラフトを作るための才能高級アパレルの満足のいく作品。[http://www.cosmopolitancarpetcleaning.com/data/images1/gaga.html �����ߥ�� �rӋ ��ǥ��`��] 特に技術革新と民間企業の発展のためのより科学的な目標の開発がより探査を行うために、 |
| | year = 1968
| | 金の宝石類の人気追求を費やして、 |
| | pages = 115–152
| | 第五（中国語）靴のために開催されているEコマース業界のサミットの登録はまだ正式にコミュニティから注目されていません。 |
| | url =
| | 日付をウェイクアップすることは困難で夢を見て：2013年8月24日10時00分46秒マンチェスター·シティは、 [http://www.equityfair.ch/mod_news/jp/mall/shoes/cl/ ���ꥹ���<br><br>�֥��� ������] |
| }}</ref> as an efficient means of solving the incompressible [[Navier-Stokes equation]]s. The key advantage of the projection method is that the computations of the velocity and the pressure fields are decoupled.
| |
|
| |
|
| ==The algorithm==
| | Visit my blog [http://citruscontrols.com/Consulting/converse.html コンバース ハイカット] |
| The algorithm of projection method is based on the [[Helmholtz decomposition]] (sometimes called Helmholtz-Hodge decomposition) of any vector field into a [[solenoidal field|solenoidal]] part and an [[irrotational field|irrotational]] part. Typically, the algorithm consists of two stages. In the first stage, an intermediate velocity that does not satisfy the incompressibility constraint is computed at each time step. In the second, the pressure is used to project the intermediate velocity onto a space of divergence-free velocity field to get the next update of velocity and pressure.
| |
| | |
| ==Helmholtz–Hodge decomposition==
| |
| The theoretical background of projection type method is the decomposition theorem of [[Olga Aleksandrovna Ladyzhenskaya|Ladyzhenskaya]] sometimes referred to as Helmholtz–Hodge Decomposition or simply as Hodge decomposition. It states that the vector field <math>\mathbf{u}</math> defined on a [[simply connected space|simply connected]] domain can be uniquely decomposed into a divergence-free ([[Solenoidal vector field|solenoidal]]) part <math>\mathbf{u}_{\text{sol}}</math> and an [[Conservative vector field#Irrotational vector fields|irrotational]] part <math>\mathbf{u}_{\text{irrot}}</math>.<ref>{{cite book | title = A Mathematical Introduction to Fluid Mechanics | author1 = Chorin, A. J. | author2 = J. E. Marsden | edition = 3rd | publisher = [[Springer Science+Business Media|Springer-Verlag]] | year = 1993 | isbn = 0-387-97918-2}}</ref> Thus,
| |
| | |
| :<math>
| |
| \mathbf{u} = \mathbf{u}_{\text{sol}} + \mathbf{u}_{\text{irrot}} = \mathbf{u}_{\text{sol}} + \nabla \phi
| |
| </math>
| |
| | |
| since <math>\nabla \times \nabla \phi = 0</math> for some scalar function, <math>\,\phi</math>. Taking the
| |
| divergence of equation yields
| |
| | |
| :<math>
| |
| \nabla\cdot \mathbf{u} = \nabla^2 \phi \qquad ( \text{since,} \; \nabla\cdot \mathbf{u}_{\text{sol}} = 0 )
| |
| </math>
| |
| | |
| This is a [[Poisson equation]] for the scalar function <math>\,\phi</math>. If the vector field <math>\mathbf{u}</math> is known, the above equation can be solved for the scalar function <math>\,\phi</math> and the divergence-free part of <math>\mathbf{u}</math> can be extracted using the relation
| |
| | |
| :<math>
| |
| \mathbf{u}_{\text{sol}} = \mathbf{u} - \nabla \phi
| |
| </math>
| |
| | |
| This is the essence of solenoidal projection method for solving incompressible
| |
| Navier–Stokes equations.
| |
| | |
| ==Chorin's projection method==
| |
| The incompressible Navier-Stokes equation (differential form of momentum equation) may be written as
| |
| :<math>
| |
| \frac {\partial \mathbf{u}} {\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} = - \frac {1}{\rho} \nabla p + \nu \nabla^2 \mathbf{u}
| |
| </math>
| |
| | |
| In Chorin's original version of the projection method
| |
| ,<ref>
| |
| {{Citation
| |
| | surname1 = Chorin
| |
| | given1 = A. J.
| |
| | title = Numerical Solution of the Navier-Stokes Equations
| |
| | journal = Math. Comp.
| |
| | volume = 22
| |
| | year = 1968
| |
| | pages = 745–762
| |
| | url =
| |
| }}</ref> the intermediate velocity, <math>\mathbf{u}^*</math>, is explicitly computed using the momentum equation ignoring the pressure gradient term:
| |
| | |
| :<math>
| |
| \quad (1) \qquad \frac {\mathbf{u}^* - \mathbf{u}^n} {\Delta t} = -(\mathbf{u}^n \cdot\nabla) \mathbf{u}^n + \nu \nabla^2
| |
| \mathbf{u}^n
| |
| </math>
| |
| | |
| where <math>\mathbf{u}^n</math> is the velocity at <math>\,n</math><sup>th</sup> time level. In the next (projection) step, we have
| |
| | |
| :<math>
| |
| \frac {\mathbf{u}^{n+1} - \mathbf{u}^*} {\Delta t} = - \frac {1}{\rho} \, \nabla p ^{n+1}
| |
| </math>
| |
| | |
| Re-writing the above equation for the velocity at <math>\,(n+1)</math> level, we have
| |
| | |
| :<math>
| |
| \quad (2) \qquad \mathbf{u}^{n+1} = \mathbf{u}^* - \frac {\Delta t}{\rho} \, \nabla p ^{n+1}
| |
| </math>
| |
| | |
| Computing the right-hand side of the above equation requires a knowledge of the pressure, <math>\,p</math>, at <math>\,(n+1)</math> level. This is obtained by taking the [[divergence]] and requiring that <math>\nabla\cdot \mathbf{u}^{n+1} = 0</math>, which is the divergence-free(continuity) condition, thereby deriving the following Poisson equation for <math>\,p^{n+1}</math>,
| |
| :<math>
| |
| \nabla ^2 p^{n+1} = \frac {\rho} {\Delta t} \, \nabla\cdot \mathbf{u}^*
| |
| </math>
| |
| It is instructive to note that, the equation written in the following way
| |
| :<math> | |
| \mathbf{u}^* = \mathbf{u}^{n+1} + \frac {\Delta t}{\rho} \, \nabla p ^{n+1}
| |
| </math>
| |
| is the standard Hodge decomposition if boundary condition for <math>\,p</math> on the domain boundary, <math>\partial \Omega</math> is <math>\nabla p^{n+1}\cdot \mathbf{n} = 0</math>.
| |
| | |
| For the explicit method, the boundary condition for <math>\mathbf{u}^*</math> in equation (1) is natural. If <math>\mathbf{u}\cdot \mathbf{n} = 0</math> on <math>\partial \Omega</math>, is prescribed, then the space of divergence-free vector field will be orthogonal to the space of irrotational vector fields, and from equation (2) one has
| |
| :<math>
| |
| \frac {\partial p^{n+1}} {\partial n} = 0 \qquad \text{on} \quad \partial \Omega
| |
| </math>
| |
| The explicit treatment of the boundary condition may be circumvented by using a [[staggered grid]] and requiring that <math>\nabla\cdot \mathbf{u}^{n+1}</math> vanish at the pressure nodes that are adjacent to the boundaries.
| |
| | |
| A distinguishing feature of Chorin's projection method is that the velocity field is forced to satisfy a discrete continuity constraint at the end of each time step.
| |
| | |
| == General method ==
| |
| | |
| Typically the projection method operates as a two-stage fractional step scheme, a method which uses multiple calculation steps for each numerical time-step. In many projection algorithms, the steps are split as follows:
| |
| | |
| # First the system is progressed in time to a mid-time-step position, solving the above transport equations for mass and momentum using a suitable advection method. This is denoted the ''predictor'' step.
| |
| # At this point an initial projection may be implemented such that the mid-time-step velocity field is enforced as divergence free.
| |
| # The ''corrector'' part of the algorithm is then progressed. These use the time-centred estimates of the velocity, density, etc. to form final time-step state.
| |
| # A final projection is then applied to enforce the divergence restraint on the velocity field. The system has now been fully updated to the new time.
| |
| | |
| == References ==
| |
| {{reflist}}
| |
| | |
| [[Category:Computational fluid dynamics]]
| |
| [[Category:Mathematical physics]]
| |
| [[Category:Computational physics]]
| |
Hi, everybody! My name is Theresa.
It is a little about myself: I live in Austria, my city of Pollach.
It's called often Eastern or cultural capital of SALZBURG. I've married 4 years ago.
I have 2 children - a son (Columbus) and the daughter (Noah). We all like Amateur radio.
xunjie 風の動きが最先端のニューヨーク、
同社は日本と西欧およびその他の国から一流のプロ子供服の生産設備を持っている以上の5台、
地元やその他の状況を支援するリーディングカンパニーによると、 [http://www.dressagetechnique.com/images/jp/top/jimmychoo/ ���ߩ`��奦 �ȩ`�ȥХå�] カナリ2012年秋と冬のシリーズは、
すべてのスタッフの出店商売繁盛におめでとうと言いたいが活況を呈しています。
糸くずのブーツのすべての種類のための時間で組織になりましたUggのブーツは、 [http://www.equityfair.ch/gzd/jr/mall/shoe/newbalance/ �˥�`�Х�� ���˩`���` ���] 例外的に魅力的なカーブを作成するのに役立つことができることを感じさせるだけではありません。
完全にマルチアングルディスプレイは英国のファッションオリエンタルスタイルに属します。
プラス私たちが作成するためのクラフトを作るための才能高級アパレルの満足のいく作品。[http://www.cosmopolitancarpetcleaning.com/data/images1/gaga.html �����ߥ�� �rӋ ��ǥ��`��] 特に技術革新と民間企業の発展のためのより科学的な目標の開発がより探査を行うために、
金の宝石類の人気追求を費やして、
第五（中国語）靴のために開催されているEコマース業界のサミットの登録はまだ正式にコミュニティから注目されていません。
日付をウェイクアップすることは困難で夢を見て：2013年8月24日10時00分46秒マンチェスター·シティは、 [http://www.equityfair.ch/mod_news/jp/mall/shoes/cl/ ���ꥹ���
�֥��� ������]
Visit my blog コンバース ハイカット