دانلود مقاله ISI انگلیسی شماره 27848
عنوان فارسی مقاله

ابزارهای تجزیه و تحلیل عملکرد کاربردی برای الگوریتم موازی نشت مرزی مش آزاد تطبیقی ​​المان محدود

کد مقاله سال انتشار مقاله انگلیسی ترجمه فارسی تعداد کلمات
27848 2005 16 صفحه PDF سفارش دهید 4740 کلمه
خرید مقاله
پس از پرداخت، فوراً می توانید مقاله را دانلود فرمایید.
عنوان انگلیسی
Performance analysis tools applied to a finite element adaptive mesh free boundary seepage parallel algorithm
منبع

Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)

Journal : Computer Methods in Applied Mechanics and Engineering, Volume 194, Issues 2–5, 4 February 2005, Pages 297–312

کلمات کلیدی
ابزارهای تجزیه و تحلیل عملکرد - المان محدود - نشت مرزی آزاد - محاسبات موازی -
پیش نمایش مقاله
پیش نمایش مقاله ابزارهای تجزیه و تحلیل عملکرد کاربردی برای  الگوریتم موازی نشت مرزی مش آزاد تطبیقی ​​المان محدود

چکیده انگلیسی

A finite element, adaptive mesh, free surface seepage parallel algorithm is studied using performance analysis tools in order to optimize its performance. The physical problem being solved is a free boundary seepage problem which is nonlinear and whose free surface is unknown a priori. A fixed domain formulation of the problem is discretized and the parallel solution algorithm is of successive over-relaxation type. During the iteration process there is message-passing of data between the processors in order to update the calculations along the interfaces of the decomposed domains. A key theoretical aspect of the approach is the application of a projection operator onto the positive solution domain. This operation has to be applied at each iteration at each computational point. The VAMPIR and PARAVER performance analysis software are used to analyze and understand the execution behavior of the parallel algorithm such as: communication patterns, processor load balance, computation versus communication ratios, timing characteristics, and processor idle time. This is all done by displays of post-mortem trace-files. Performance bottlenecks can easily be identified at the appropriate level of detail. This will numerically be demonstrated using example test data and comparisons of software capabilities that will be made using the Blue Horizon parallel computer at the San Diego Supercomputer Center.

مقدمه انگلیسی

The problem studied is the free surface seepage problem shown in Fig. 1. The following assumptions are made: the soil in the flowfield is homogeneous and isotropic; capillary and evaporation effects are neglected; the flow obeys Darcy’s Law; the flow is two-dimensional and at steady state. Because of the assumptions made, the problem is described by the velocity potential function, ϕ , whose governing differential equation and boundary conditions are also shown in Fig. 1. The relevant dimensions are taken to be: x 1 = 40, y 1 = 10 and y 2 = 3. In Fig. 1, Ω is the seepage region abdf. The location of the curve fd , View the MathML sourcey=f¯(x), is unknown a priori. Full-size image (6 K) Fig. 1. The example physical problem (free boundary seepage). Figure options A fixed domain formulation for this problem can be obtained by using the Baiocchi method and transformation (see [2], [10], [11], [3] and [6]). In this approach the a priori unknown solution region is extended across the free surface into a known region. The dependent variable is also continuously, similarly extended. Then a new dependent variable is defined using Baiocchi’s transformation within these regions. The resulting problem formulation leads to a ‘complementarity system’ associated with its respective variational inequality formulation. This method has proven effective not only from the purely theoretical point of view, but also from the point of view of yielding new, simple, and efficient numerical solution schemes. Fig. 2 shows the governing equations and boundary conditions that describe the fixed domain formulation of the problem presented in Fig. 1. D is the region abef. The variable w is the Baiocchi transformation of the extended potential function, i.e., equation(1) View the MathML sourcew(x,y)=∫yy1ϕ˜(x,η¯)-η¯dη¯, Turn MathJax on where equation(2) View the MathML sourceϕ˜(x,y)=ϕ(x,y)inΩ¯,ϕ˜(x,y)=yinD¯-Ω¯. Turn MathJax on The detailed derivations of these equations are given in [3]. Full-size image (6 K) Fig. 2. The example physical problem (free boundary seepage) for numerical implementation. Figure options The problem shown in Fig. 2 can be written as a ‘complementary system’ and its corresponding variational inequality formulation. Then the following theorem can be stated: Let View the MathML sourcep¯(x,y) be the Dirichlet data in Fig. 2 and define View the MathML sourceK={v(x,y)|v∈H1(D),v|∂D=p¯,v⩾0a.e.onD}, Turn MathJax on a closed convex set, K ⊂ H1(D). Theorem 1. If w ∈ K satisfies the governing equations and boundary conditions shown in Fig. 2, then it also satisfies the variational inequality: equation(3) View the MathML sourcea(w,v-w)⩾L(v-w)∀v∈K, Turn MathJax on where equation(4) View the MathML sourcea(w,v-w)=∫∫D∇w·∇(v-w)dxdy=∫∫Dwx(vx-wx)+wy(vy-wy)dxdy Turn MathJax on and equation(5) View the MathML sourceL(v-w)=-∫∫D(v-w)dxdy. Turn MathJax on The finding of w ∈ K is equivalent to solving the minimization problem equation(6) View the MathML sourceJ(w)⩽J(v)∀v∈K, Turn MathJax on where equation(7) J(v)=a(v,v)+2(f,v)J(v)=a(v,v)+2(f,v) Turn MathJax on in which a(v, v) is a bilinear form, continuous, symmetric, positive definite on R and f ∈ R, i.e., equation(8) View the MathML sourcea(v,v)=∫∫D∇v·∇vdxdy, Turn MathJax on equation(9) View the MathML source(f,v)=∫∫Dfvdxdy. Turn MathJax on For this example problem, f = 1. The functional J has one and only one minimum in a closed convex set. The minimum is found using the following finite element algorithm: equation(10) View the MathML sourceui(n+1/2)=-1aii∑j=1i-1aijuj(n+1)+∑j=i+1Naijuj(n)+fi, Turn MathJax on equation(11) View the MathML sourceui(n+1)=Piui(n)+ωui(n+1/2)-ui(n)=max0,ui(n)+ωui(n+1/2)-ui(n), Turn MathJax on where aij = a(Ni, Nj), fi = (f, Ni), Ni is the canonical basis of RN, Pi is the projection on the convex set, i = 1, … , N, N is the number of nodal points and ω is the relaxation factor. The relaxation factor is determined empirically. Linear triangular elements will be used in the discretization. It should be noted that the projection operation in the numerical scheme must be applied during the iteration process. It cannot be applied after the iteration process has been completed since if it were, an incorrect solution would be obtained.

نتیجه گیری انگلیسی

A significant factor that affects the performance of a parallel application is the balance between communication and workload. To fully understand the performance behavior of such applications, analysis and visualization tools are needed. Two such tools, VAMPIR and PARAVER, were used to analyze the performance of the seepage application. It was seen that optimization of the parallel code can be carried out in an iterative process involving these tools to investigate performance issues.

خرید مقاله
پس از پرداخت، فوراً می توانید مقاله را دانلود فرمایید.