1 Introduction

Improvement Cache Efficiency of Explicit Finite Element Procedure and its Application to Parallel Casting Solidification Simulation



R. Tavakoli


Material Science and Engineering Department, Sharif University of Technology, P.O. Box 11365-9466, Tehran, Iran, email: tav@mehr.sharif.edu, rohtav@gmail.com


April 2006



Abstract

A simple method for improving cache efficiency of serial and parallel explicit finite procedure with application to casting solidification simulation over three-dimensional complex geometries is presented. The method is based on division of the global data to smaller blocks and treating each block independently from others at each time step. A novel parallel finite element algorithm for non-overlapped element-base decomposed domain is presented for implementation of serial and parallel version of the presented method. Effect of mesh reordering on the efficiency is also investigated. A simple algorithm is presented for high quality decomposition of decoupled global mesh. Our result shows 10-20 % performance improvement by mesh reordering and 1.2-2.2 speedup with application of the presented cache efficient algorithm (for serial and parallel versions). Also the presented parallel solver (without cache-efficient feature) shows nearly linear speedup on the traditional Ethernet networked Linux cluster.

Keywords: cache , casting simulation, data locality, domain decomposition, finite element, parallel, solidification.

1 Introduction

Finite element analysis
Many physical phenomena such as transient heat transfer and fluid flow are governed by parabolic partial differential equations (PDEs). Finite element method (FEM) (Zienkiewicz and Taylor, 2000) is one of the powerful numerical techniques for solution of PDEs. In FEM the spatial domain is discretized with arbitrary unstructured grid, and the temporal domain is usually discretized with explicit or implicit finite difference method. The explicit time integration method has certain advantages in contrast with implicit method, such as: simplicity of implementation, smaller memory requirement and better parallelism and scalability. However, they suffer the severely restricted time step size from stability requirement. In implicit time integration method, there is not any time step restriction. But in each time step one has to solve a global linear or non-linear algebraic system of equations that is severely memory and time consuming procedure and difficult to implementation specially for distributed memory architecture.

Selection between explicit and implicit time integration methods from efficiency viewpoint is not easy task and is function of several parameters such as: behavior of PDEs, size of problem and available computational resource. Generally it can be said that for highly non-linear problems the explicit method is preferable because usually large time increment leads to increasing overall computational cost, due to increasing number of non-linear iterations. Also, in some problems there are additional time step limitation criteria that don’t permit application of the large time increment. For example application of large time step leads to interface tracking inaccuracy or instability in moving boundaries and impact problems and missing latent heat effect (and consequently energy unbalance) in the phase change problems.

Cache efficient algorithm: Background
Years ago, knowing how many floating-point operations (FLOPS) were used in a given algorithm (or code) provided a good measure of the running time of a program. This was a good measure for comparing different algorithms to solve the same problem. This is no longer true (Douglas et al., 2000a).

The growing speed gap between processor and memory has led to the development of memory hierarchies and to the widespread use of caches in modern processors (Sellappa and Chatterjee, 2004). The highest level is the most expensive, smallest and fastest of all layers. Lower levels are progressively larger, cheaper and slower. The layers (from highest to lowest) are registers, cache and main memory. The cache is itself a hierarchy L1, L2, … (levels of cache). One or two levels of cache is typical. The L1 cache is smaller and 2 to 6 times faster than the L2 cache. The L2 cache in turn is smaller than but still 10 to 40 times faster than main memory. The data held in any level of cache is usually a subset of the next larger level. The smallest block of data moved in and out of a cache is a cache line. A cache line holds data, which is stored contiguously in main memory. If the data that the CPU requests are in a cache line, it is called a cache hit. Otherwise data must be copied from a lower level of memory into a line, a cache miss. The hit rate is the ratio of cache hits to total CPU requests. The miss rate is 1 minus the hit rate. It is clearly advantageous to maximize the number of cache hits, since cache is faster than main memory in fulfilling the CPU’s request (Hu, 2000).

Therefore the speed of a code depends increasingly on how well the cache structure is exploited and the number of cache misses provides a better measure for comparing algorithms than the number of FLOPS. Unfortunately, estimating cache misses is difficult to model a priori and only somewhat easier to do a posteriori (Douglas et al., 2000a). When a program references a memory location, the data in the referenced location and nearby locations are brought into a cache. Any additional references to data before these are evicted from the cache will be one or two orders of magnitude faster than references to main memory. So increasing data locality leads to decreasing cache miss. A program is said to have good data locality if, most of the time, the computer finds the data referenced by the program in its cache. The locality of a program can be improved by changing the order of computation and/or the assignment of data to memory locations so that references to the same or nearby locations occur near to each other in time during the execution of the program ( Strout et al., 2004).

When we deal with finite element analysis (FEA) on unstructured mesh, the data connectivity has unstructured sparse structure and data storage must involve indirect addressing. So the cache miss is considerably increased due to the irregularity of many of the memory accesses. Many researchers have explored the benefits of reordering sparse matrices resulted from FEA (Pinar and Heath,1999; Im and Yelick, 1999; Oliker et al., 2000; Lohner, 2001,). In this method the bandwidth of global matrix (resulted from FEA) is reduced with node renumbering which decreases chance of cache miss during computation. But this method shows little benefit for large-scale problems on uniprocessors (Im and Yelick, 1999; see also experimental results section).

Related works in the category of cache efficient numerical algorithm were focused to improvement cache efficiency of solution of system of linear equations with stationary iterative methods (e.g. Gauss-Seidel as direct solver or as a smoother in the multigrid methods). So related works fall in the category of cache efficient implicit FEA.

Work related to performance transformations for stationary iterative methods can be categorized by whether it deals with regular or irregular meshes and whether it attempts to improve intra-iteration locality and/or inter-iteration locality. Another important distinction is between code transformations that have been automated in a compiler versus programming transformations that require some domain-specific knowledge and are currently applied by hand.

Tiling is the process of decomposing a computation into smaller blocks and doing all of the computing in each block one at a time. Tiling is an attractive method for improving data locality. In some cases, compilers can do this automatically. However, this is rarely the case for realistic scientific codes. In fact, even for simple examples, manual help from the programmers is, unfortunately, necessary (Douglas et al., 2000b).

Traditional tiling (Wolfe, 1987; Irigoin and Triolet, 1988; Gannon et al., 1988; Wolfe, 1989; Wolf and Lam, 1991; Carr and Kennedy, 1992; McKinley et al., 1996) can be applied to a perfect loop nest that traverses the unknowns associated with a regular mesh, provided the memory references and loop boundaries are affine and the unknowns are ordered in a way that allows a compile-time analysis to find a legal tiling. In particular, after the enabling transformation skewing is applied, tiling is often applicable to Gauss-Seidel and SOR over a regular mesh. For Jacobi over a regular mesh, tiling transformations developed for imperfectly nested loops can be used (Song and Li, 1999; Ahmed et al., 2000). Another issue involved in tiling computations on regular meshes is how to determine the tiling and array padding parameters (Rivera and Tseng, 2000). If other compiler transformations, such as skewing, function inlining and converting ’while’ loops to ’for’ loops, are used, then it is possible to apply tiling transformations for imperfectly nested loops to stationary iterative methods to achieve inter-iteration locality on regular meshes.

Increasing inter-iteration locality through programming transformations for iterative stationary methods on structured meshes is explored by Leiserson et al. (1997) with blocking covers technique, Stals and Rüde (1997) with cache blocking along one dimension for two-dimensional problem, Kodukula et al. (1997) with data-centric multi-level blocking technique, Bassetti et al. (1998) with stencil optimization techniques, Bassetti et al. (1999) with combining blocking covers with stencil optimization techniques, Povitsky (1999) with Wavefront cache-friendly algorithm, Pugh and Rosser (1999) with iteration space slicing, Wonnacott (2002) with time skewing, Douglas et al. (2000a) and Sellappa and Chatterjee (2004) with two-dimensional cache blocking techniques. Such programming transformations are explored extensively due to the prevalence of iterative regular mesh computations that compilers do not tile because the necessary combinations of enabling transformations are not done automatically. For unstructured grids, Cache blocking (Douglas et al., 2000) and sparse tiling(Strout et al., 2004) are two such run-time reordering transformations developed for the stationary iterative methods. Im (2000), Im and Yelick (2001) and Im et al. (2004) have developed a code generator, SPARSITY, that improves the intra-iteration locality for the general sparse matrix operations by register blocking and cache blocking techniques.

There has also been work on compiler-generated inspectors/executors for improving intra-iteration locality of irregular problems (Ding and Kennedy, 1999; Mitchell et al., 1999; Han and Tseng, 2000; Mellor-Crummey et al., 2001). These papers describe how a compiler can analyze non-affine array references in a loop and generate the inspectors and executors for performing run-time data and iteration reordering. These transformations can be applied to the inner loops of Jacobi implemented for sparse matrix formats, but not to Gauss-Seidel or SOR due to the intra-iteration data dependences.

Outline of present work
As stated in the above section most of works in the category of cache efficient algorithms have been attended to implicit and iterative methods. So extension of those ideas to explicit and non-iterative methods (for developing cache efficient algorithms) can be valuable work, which is a part of the present study.

The scope of the present study is high performance implementation of explicit finite element procedure for solution of large-scale problems, which are governed with non-linear parabolic PDEs. In this study our focus is on the simulation casting solidification, which is a non-linear phase change problem. For this purpose at first step we try to increase cache efficiency of explicit finite element procedure for single processor execution and then extend it to distributed memory parallel execution. Although we focus on the special problem but the introduced idea is general and extendable to other applications.

The original contributions of this paper are: (i) Increasing cache efficiency of the fully explicit finite element procedure by decomposition of the global data to smaller sub-blocks with application of domain decomposition concept. (ii) Extension of the introduced idea to parallel processing by application of secondary inter-processor domain decomposition procedure. (iii) Introducing easy to implement parallel explicit finite element procedure for solution of related PDEs on the non-overlapped element-base decomposed domain. (iv) Solution of some technical problems behind suitable domain decomposition for practical casting simulation (problems due to presence of decupled meshes, e.g., cast mesh is completely decupled from mold mesh). (v) Application of high performance computation for solution of a practical engineering problem, i.e., casting solidification simulation.

The rest of this paper is organized as follows. In the next section, the governing equations are presented. Section 3 contains the numerical finite element approximation of the governing equations. In Section 4 we present our cache efficient solver and details of its implementation for serial and parallel execution. Section 5 is devoted to experimental result and section 6 summarizes the present work.

2 Mathematical modeling

From a macroscopic point of view, if effect of melt flow during solidification is neglected, solidification is governed by the heat conduction equation. The appropriate mathematical description of the heat conduction process in a stationary material region, is given by,

(1)

where is the temperature field, is the density, is the specific heat and is the thermal conductivity. The initial condition is

(2)

where is pouring temperature of liquid metal. The boundary condition used are given by:

(3)
(4)

where , and are specified boundary heat flux, convective heat transfer coefficient and the ambient temperature respectively and is the direction of the unit normal vector to . and are the portions of the boundary of the computational domain .

The enthalpy method is one of the popular methods for incorporation of phase change effect in the conductive heat transfer equation. in this manner the equation (1) has the following form (Swaminathan and Voller, 1992):

(5)

where is the enthalpy field. In general the enthalpy could be a function of a number of variables, such as temperature, concentration, cooling rate, etc. In many solidification model, however, the enthalpy in the mushy region can be assumed to be a function of temperature alone. There are several method for solution of equation (5) that, in the present study the apparent heat capacity method (Comini et al., 1974) is used as follow:

(6)
(7)

in equation (6) called apparent heat capacity, that is evaluated by Lemmon (Lemmon, 1981) method in this study. This method shows good robustness and accuracy (Thomas et al., 1984).

An essential step in the accurate simulation of the cooling of a cast part is to model the effect of air gap formation between the mould and casting and other contact resistance between two components of casting system. The problem of contact resistance between two stationary materials may be represented by either of two methods. One approach to modeling this effect assumes that the contact region is composed of a fictitious material, of small thickness, whose properties produce the appropriate resistance to heat flow across the interface. Typically, this gap material will have a negligible heat capacity and a nonlinear conductivity. A more convenient model can be introduced if the interface contact is seen as a convective thermal resistance between two surfaces. In the other words, one can easily replace the concept of contact conductance with an equivalent convection heat transfer coefficient. The model can also accommodate radiation heat transfer between surfaces of the mould and casting. In this model the geometry, including the casting and mould, is separated by an interface boundary line in the two-dimensional and a surface in the three-dimensional case. The interface boundary is subdivided into a number of segments that represent different interface conditions. Each boundary segment is identified by two interface segments; a segment which belongs to the casting region and a corresponding segment associated with the mould region. These two segments are actually lying on each other and represent the same boundary. For each boundary segment in the casting, the corresponding boundary segment in the mould acts as an ambient condition and vice versa. This means that the heat transfer between these two surfaces is characterized by their temperatures and the local heat transfer coefficient specified for that segment of the interface:

(8)

where is local heat transfer coefficient of interface. the value of must be extracted from material data base. It can be fixed or be function of time and temperature. The concept of interfacial convective heat transfer can easily be applied to any kind of contact surface in casting configuration. This might be a surface between a chill and casting or mould and a surface separating two parts of a mould as shown in Figure 1. For instance, and represent interfacial boundaries of the cast and cope for segment . In this way, virtually all contact surfaces in the casting system can be conveniently modeled (Manzari et al., 2000). The other benefits of this approach are:

  1. It is possible to generate mesh separately in cast and mold and don’t need matching element in the interface (see Figure 2). This gives more flexibility to mesh generator and lead to better mesh quality and also increase mesh generation efficiency.

  2. The element sizes for two sides of the interface can be significantly different. This is often useful as a relatively dense mesh is normally employed for the casting region in comparison with the mould region.

  3. There is no need to define the thermal conductivity and air gap thickness as they are uniquely replaced by the interfacial heat transfer coefficient.

  4. Any standard finite element program can be used for this purpose and no modification is required.

  5. This model will help to assure a smoother temperature distribution when starting with a large temperature difference between two sides of the interface.

  6. The ability to accommodate both spatial and temporal variations in the interfacial heat transfer coefficients.

For more detail about the solidification modeling see Refs. Huy and Argyropoulos (1996) and Lewis and Ravindran (2000).

Figure 1: Separation of regions at interfaces and assignment of the associated boundary conditions (this figure is adapted from a presentation by Prof. Mehrdad T. Manzari).
Figure 2: Schematic of non-matching elements coordination for a typical cast-mold interface.

3 Numerical modeling

For solution of the governing equations we subdivide spatial physical domain with unstructured four node tetrahedral elements. The standard Galerkin form of the weighted residual method is used to discretize equation (6) in space by the FEM, which yields the following non-linear system first order differential equations a system of first-order ordinary differential equations:

(9)

where and are respectively capacitance conductance matrices, vector T is the vector of nodal unknowns and the right–hand side vector F contains boundary fluxes. The mass lumping method (diagonalizing capacitance matrix with ’row-sum’ technique) is used for calculation of capacitance matrix. It has better stability and computational efficiency and also in our parallel implementation needs lower communications cost. The traditional finite difference method is generally used to obtain a fully discretised algebraic system of equations for the time domain as follows (Lewis et al., 1996):

(10)

where is time step and is blending factor for switching between explicit and implicit method. First order fully explicit and implicit method is achieved with equal to 0 and 1 respectively and give implicit second order accurate Crank-Nicholson method. The nodal values of enthalpy are related to the temperature by:

(11)

where is the enthalpy per unit mass (variation of density is neglected in this study). The temperature field is then obtained by solving such a set of equations. Because of the apparent heat capacity and thermal conductivity are function of temperature field, equation (9) is non-linear. In this study, equation (9) is linearized by using known values of apparent heat capacity and thermal conductivity (from previous step) and done iterative updating procedure until achieving convergence as follows:

(12)

where superscript is related to non-linear iteration. After convergence of non-linear iteration, last values of temperature field is used as . In the explicit time integration method, by using enough small time step which, is usually stability bound of explicit method, non-linear loop is not essential (for more detail see Lewis et al. (1996) and Reddy and Gartling (2000)). The stability bound of the explicit time integration method is approximately given by,

(13)

where is density, is the nodal heat capacity, is the nodal heat conductivity and is the minimum mesh distance. Although the implicit time integration method is unconditionally stable but increasing time step size leads to decreasing overall time accuracy and slow convergence rate (or divergence) of non-linear iterations.

As mentioned in previous section the heat transfer between cast and mold is replaced by equivalent convective heat flux. The ambient temperature is found for each interface element by using the average temperature of the its closest element in the other region. Therfore at start stage of simulation closest element of each interface element (in the other region) is found and is labeled as sister of it for interpolation of ambient temperature during simulation.

4 Implementation of cache efficient solver

Since the parallel implementation of solver is a part of this study and it has complete analogy with implementation of cache efficient algorithm we present method of parallel implementation first and then describe our cache efficient solver.

4.1 Parallel implementation

4.1.1 Domain decomposition

The first aspect of parallelizing a finite element method by domain decomposition is to divide the grid into parts. This is how the full computational task is divided among the various processors of the parallel machine. Each processor works only on its specific portion of the grid and anytime a processor needs information from another processor a message is passed.

For the best parallel performance, one would like to have optimal load balancing and as little communication between processors as possible. Consider load balancing first. Ideally, one would like each processor to do exactly the same amount of work every iteration. That way, each processor is always working and not sitting idle. For a finite element code, the basic computational unit usually is the element. It makes sense to partition the grid such that each processor gets an equal (or nearly equal) number of elements to work on. The second criterion is that the amount of communication between processors be made as small as possible. This needs minimizing number of edge cuts and the degree of adjacency (number of neighbors) for each processor. For partitioning procedure we use the METIS library (Karypis and Kumar, 1998; Karypis and Kumar, 2005) that done first criteria excellently and second criteria in the acceptable manner.

As mentioned in previous section, mesh generation of each component of casting configuration is done separately and in contact surfaces there aren’t essentially coincidence of grid points. Therefore we have two (or more) separated grids. This discontinuity in global grid, decrease quality of mesh partitioning, i.e., increase probability of formation of low quality and separated sub-domains. It is natural result when the two or more meshes without any physical connection are undergone domain decomposition simultaneously.

This problem is solved in this study by adding some virtual elements between each two contact surfaces, i.e., for each cast surface element the corresponding sister element is indicated and by its three nodes and one nodes of its sister element, a virtual tetrahedron is formed and added to end of element connectivity list (see Figure 3). These virtual elements are only used for domain decomposition purpose. For improvement of this algorithm it is better to add such virtual elements only in one side of two regions for decreasing the number of sub-domains that are shared in two regions (cast and mold). Figure 4 shows the simple casting system that has two separated mesh (cast and mold) which is divided to six sub-domains by METIS library. The left section of Figure 4 shows result of traditional partitioning and right section, shows result of presented algorithm (adding virtual element to left side of cast-mold interface). The number of edge cuts for this sample are: 6539 and 1625, for traditional partitioning and presented algorithm respectively. Also the load balancing of the presented domain decomposition method is about better than the primary method. It can be seen that by using presented algorithm, the quality of partitioning procedure is increased considerably.

The non-overlapped element based domain decomposition method is used for parallelizing finite element procedure in the present study. In this manner the elements are assigned uniquely to partitions. Therefore after domain decomposition each processor has own private nodes and shared nodes with other processors in its boundaries with other sub-domains. If boundary between two neighbor sub-domains is cast-mold interface (or in general case interface of two casting components), we have additional kind of nodes that we called them as external nodes (nodes related to sister elements) as shown in Figure 5.

Figure 3: Schematic of addition of virtual element to cast-mold interface.
Figure 4: Comparison between result of traditional domain decomposition (left) and adding virtual contact element (right).

4.1.2 Parallel implementation of explicit solver

When in equation (10), the pure explicit algorithm is preserved as follows:

(14)

For calculation of new temperature field, the global capacitance and conductance matrix must be calculated. Because of we use mass lumping method matrix is diagonal, therefore updating procedure of each node contains one matrix-vector product and two vector summation and one vector scaling.

In a distributed environment, calculation of and expression requires communication between processors. The capacitance matrix, (is diagonal), the conductivity matrix, , and boundary flux vector, can be written as

(15)
(16)
(17)

where , and are the contribution from the element and is the total number of elements. and represent the local element contribution ( and respectively) expanded to system size, i.e., the entries of are mapped into corresponding global row and column locations with others entries in equal to zero. Then equation (14) can be written as

(18)

is a very sparse matrix, but only the dense local element contribution needs to be stored. and can be calculated concurrently and independently for elements . We define the local capacitance matrix ( ), the local conductivity matrix ( ) and the local boundary flux vector ( ) of each processor as follows

(19)
(20)
(21)

where is the number of element in each partition (related to each processor). So we have

(22)

where is number of partitions and is local temperature vector of ith-partition. Note that each partition must have the global values of vector in its shared nodes for performing above procedure.

Updating procedure in this study is as follows: Using traditional element-by-element finite element procedure and calculation of and expression , locally within each partition. For internal nodes the above procedure is complete and equation (4.1.2) can be used for updating temperature field. But any shared node, needs effects of elements that constructed from it, but posed in the neighbor partitions. These effects are equal to calculated values of and expression by neighbor partitions. Therefore neighbor partitions must exchange these calculated values related to their shared nodes and perform summation on exchanged data. Note that for imposition of boundary condition each block needs to temperature values of its external nodes (see Figure 5) that achieves them via communication. After this step, new temperature field is calculated with equation (4.1.2).

The size of data that must be communicated between two neighbor processors is equal to, two times of number of their shared nodes. The partitions that have mold-cast interface as inter partition boundary, need additional communication for obtaining temperature of theire external nodes.

Note that main difference of our parallel solver with related works are non-overlapped domain decomposition and communication of calculated values vs. overlapped domain decomposition and communication of temperature values of overlapped region.

Figure 5: Element based domain decomposition and definition of various kind of nodes that are used in this study.

4.1.3 Communication scheduling

We need for a communication schedule to coordinate the exchange of information between processors. A communication schedule simply tells each processor when and with whom it can exchange its message. This is necessary because a processor can only communicate with one other processor at a time.

Consider the communication graph in Figure. 6. Each node represent one processor and each edge shows needful communication. In this case there are 4 separate communication stages, and in all but the third, two separate pairs of processors swap information at the same time. For example, in stage 2, processors 1 and 3 swap as do processors 2 and 0. Processor 4 is idle. So you can see that it isn’t necessary to have a separate communication stage for each pair of processors that need to communicate. It is only necessary to make sure that pairs of processors swap without conflict.

This communication schedule is constructed by coloring the edges of the graph so that each edge coming from a vertex has a different color. An attempt is made to minimize the number of colors used which equals the number of communication stages that must be performed. Here is an outline of the algorithm:

  1. Determine the vertex of highest degree. Call it the root

  2. Starting at the root look at all the edges emanating from the vertex and determine which ones have not yet been colored.

  3. Mark all these uncolored edges with a different color starting with the lowest available color and working up.

  4. Do this same procedure for all the rest of the vertices in the graph until all the edges are colored.

There is a theorem in graph theory called Vizings theorem which states that the most colors one would ever have to use for this is equal to the degree of the root plus one. The algorithm presented above has never produced more colors than dictated by Vizings theorem.

Figure 6: A sample of communication graph.

4.1.4 Message passing interface

We use MPI library for the message passing implementation of the exchange algorithm. In this section we describe simple blocking communication method that is used in this study.

Using blocking massage passing needs care for prevention of decaying efficiency and system dead-lock. Scheduling communication table (that is described in previous section) is essential for efficient implementation of blocking communication (although with non-blocking communication, scheduling is not essential, but fore controlling network traffic and consequently achieving better performance scheduling is recommended). For this purpose we perform communication based on table of communication with function MPI_Sendrecv. This call performs a data swap, by combining the send and receive into a single call. The communication with this routine is faster than doing a separate synchronous send and synchronous receive. The other benefit is prevention of system dead lock that may be happen during blocking communication.

4.2 Improvement cache efficiency

As stated in the introduction section, the key idea behind optimization of cache behavior is improvement data locality with decomposition of computation into smaller parts (cache blocks) and doing all of the computing in each cache block one at a time. For this purpose we use from domain decomposition concept. So we decompose the global mesh to appropriate number of cache blocks and treat each block independently from others and anytime a block needs information from another blocks communicate (virtually) with them. This algorithm is similar with our parallel implementation. The main difference is that in the parallel solver the communication between neighbor partitions is performed via network (intra-processor communication), while in the cache efficient algorithm, communication between neighbor blocks is performed directly (inter-processor virtual communication). Therefore the domain decomposition, solver implementation and essential data structure are same as parallel solver.

This method can be easily extended to parallel processing. For this purpose, after primary domain decomposition based on load balancing criteria and division of global mesh between processors, each processor performs the secondary domain decomposition for construction of its cache blocks. Figure 7 shows schematically this procedure.

Optimal number of cache block is severely function of cache size, specially L2-cache. In the present study this parameter is indicated with simple numerical experiment. For this purpose each processor changes number of its blocks and solve problem locally for few number of time steps (without communication with other processors). Then the optimal number of blocks for each processor is indicated based on the measured elapsed CPU time. As we deal with transient problem the computational overhead of this stage is negligible (below 1 % of total time for practical application in our experiment).

The other advantage of this algorithm (which is not in the scope of present study) is providing possibility of multi-thread execution on each processor (executing computation of each block or each some blocks as one thread). This is very important for high performance computation when we deal with hyper-threaded technology CPUs or dual-processor CPUs. For such systems single thread execution can take only 50 % of CPU’s power in the extreme case. But with multi-threading it is possible to take nearly 100 % of CPU usage.

Figure 7: Two level domain decomposition method: original domain is first partitioned based on load balancing criteria and divided between processors, and then each processor divides its local mesh to appropriate number of block for improving cache efficiency, this figure is adapted from (Gullerud and Dodds).

5 Result and disscution

In this section we investigate performance of presented algorithm by solution of the problem for six test cases with various mesh resolution and configuration. In these cases grids are generated individually over casting and mold part without attention to matching grids at cast-mold interfaces. Figure 8 shows the surface mesh (cases 1-4) and Table 2 shows the mesh properties of these cases.

Figure 8: Surface mesh configuration of cases 1-4 in this study.

5.1 Computational resource

As the computing platform for parallel execution, we have used a 14 nodes Beowulf cluster based on 900 MHz Intel Pentium III processors and switched Ethernet networks (100Mbps). Table 1 shows specification of various machines which were used for serial execution in the present study.

5.2 Performance of mesh reordering

In this experiment we investigate effect of mesh reordering with reverse Cuthill-McKee method (Alan and Liu, 1981) on performance of serial code. Figure 9 shows the sparsity pattern of global coefficient matrix befor and after mesh reordering and Table 3 shows the effect of reordering on the bandwidth of global coefficient matrix. It is clear that the bandwidth of global coefficient matrix is reduced considerably with reordering procedure. Table 4 shows the effect of reordering on performance of sequential explicit finite element procedure for cases 1-6 on various machines. It can be seen that the performance of computation is increased with this procedure and this effect is increased with decreasing size of the global data.

5.3 Performance of the serial cache efficient algorithm

In this section we present results of application of the presented method on the efficiency of sequential explicit finite element procedure. We define the performance improvement factor as relation of CPU time of original explicit finite element procedure to CPU time of presented cache efficient finite element procedure.

Figures 10 - 15 show the variation of improvement factor with variation of number of blocks for cases 1-6 for non-reordered (a) and reverse Cuthill-McKee reordered (b) mesh. Plots show that with increasing number of blocks the performance is increased and a performance peak is achieved in the appropriate number of blocks. The performance improvement is generally better for the large meshes. This is natural result because with increasing size of data the probability of cache miss is increased due to increasing data sparsity. Also it seems that for relatively small data (cases 1-3) the performance peak of reordered mesh is lower than non-reordered mesh. But in spite of this smaller performance peak in the all of our experiment the CPU time related to reordered mesh is generally better than non-reordered mesh. So combination of reordering and cache blocking leads to better performance. Note that with increasing number of blocks the total number of nodes of global data is increased considerably (due to presence of shared nodes) as illustrated in Figures 10 (c) - 15 (c). It is equivalent to increasing size of the global data. But in spite of this the overall effect leads to better performance (total number of elements is fixed). The average size of blocks are also plotted in 11 (d) - 16 (d).

Figure 16 shows the average block size of cases 1-6 at the performance peak for each machine. Plot shows that the optimal block size is approximately equal to size of L2-cache (except case 1 that its data is small and increasing number of blocks severely increase the size of global data due to presence of shared nodes).

5.4 Performance of the parallel cache efficient algorithm

In this section we present result of parallel version of the presented method. Figure 17 shows the speedup curves (for cases 1-6) related to parallel execution of the presented method (original and cache efficient). Note that the CPU time of serial execution of the traditional method was used as reference CPU time for calculation of speedup. Plot shows that for original method good parallel efficiency and scalability is resulted and in spite of application of the Ethernet network, nearly linear speedup is achieved for relatively heavy meshes (cases 3-6). Also the presented cache efficient algorithm preserves its efficiency in the parallel execution and the efficiency is improved with factor 1.2-2.2 (except case 1).

6 Conclusions

In this paper, a simple method is presented for improvement cache efficiency of the explicit finite element procedure with application to casting solidification simulation (non-liner heat transfer problem with phase change) over three-dimensional complex geometries. In this method the global data is divided to smaller blocks with traditional domain decomposition method and calculation related to each block is performed independently from others. A novel algorithm is presented for solution of the governing equations on the non-overlapped element-base decomposed domain. Then the presented method is extended to parallel processing without considerable additional effort. The other covered features are: presentation of a simple algorithm for improvement quality of the decomposed domain when the original global mesh contains two or more decoupled meshes; investigation effect of mesh reordering on the performance of the explicit finite element procedure; proposing efficient inter-processor data communication based on communication scheduling and blocking-mode massage passing.

Our results show that: 10-20 % performance improvement is achieved with mesh reordering; achieving nearly linear speedup with combining of the presented parallel finite element procedure and data communication method on a Ethernet networked distributed memory parallel machine; application of the presented method leads to increasing computational efficiency of serial and parallel execution with factor 1.2-2.2; the optimal block size is approximately equal to size of the L2-cache.

Acknowledgements
This article was the term project (number 2) of my parallel processing course due to Prof. Mehrdad T. Manzari. I would like to thanks Dr. M.T. Manzari for helpful discussion and sharing his sequential FEM code with me to parallelize.

Machine CPU (MHz) L1-cache (Kb) L2-cache (Kb) RAM (Mb)
Machine 1 AMD 2000 128 512 512
Machine 2 AMD 1000 128 64 256
Machine 3 Intel 2400 8 512 512
Machine 4 Intel 3200 15 1024 512
Machine 5 Intel 3000 8 512 512
Table 1: Specification of various machines, which were used for serial execution in the present study.
Case Number of Nodes Number of Tetraheders
 1 348 7638
 2 4065 18388
 3 14800 64989
 4 29205 131582
 5 109119 540280
 6 149657 741690
Table 2: Mesh properties for cases 1-6 in the present study.
Case Initial Bandwidth Bandwidth after reordering
 1 412 266
 2 2135 254
 3 14694 2285
 4 27631 4264
 5 108610 1791
 6 148658 2358
Table 3: Bandwidth of global coefficient matrix for cases 1-6 in the present study, before and after mesh reordering with reverse Cuthill-McKee method.
Case Machine 1 Machine 2 Machine 3 Machine 4 Machine 5
1 7.6 % 45.8% 5.3% 60.5% 37.1%
2 1.2% 42.2% 3.7% 48.9 % 36.1%
3 3.5% 33.5% 1.3% 6.9% 1.5%
4 1.% 30.3% 1.8% 18.2% 1.%
5 2.3% 21.7% 1.9% 10.0% 4.2%
6 3.0% 17.4% 5.1% 11.5% 7.9%
Table 4: Effect of reverse Cuthill-McKee mesh reordering on performance of sequential finite element procedure (performance improvement in percent).
Figure 9: sparsity pattern of global coefficient matrix for cases 1-6 in the present study, before (left) and after reordering with reverse Cuthill-McKee reordering method (right).
Figure 10: Effect of variation of number of blocks on (a) performance improvement factor of original mesh, (b) performance improvement factor of reordered mesh, (c) total number of nodes and (d) averaged block size for case 1.
Figure 11: Effect of variation of number of blocks on (a) performance improvement factor of original mesh, (b) performance improvement factor of reordered mesh, (c) total number of nodes and (d) averaged block size for case 2.
Figure 12: Effect of variation of number of blocks on (a) performance improvement factor of original mesh, (b) performance improvement factor of reordered mesh, (c) total number of nodes and (d) averaged block size for case 3.
Figure 13: Effect of variation of number of blocks on (a) performance improvement factor of original mesh, (b) performance improvement factor of reordered mesh, (c) total number of nodes and (d) averaged block size for case 4.
Figure 14: Effect of variation of number of blocks on (a) performance improvement factor of original mesh, (b) performance improvement factor of reordered mesh, (c) total number of nodes and (d) averaged block size for case 5.
Figure 15: Effect of variation of number of blocks on (a) performance improvement factor of original mesh, (b) performance improvement factor of reordered mesh, (c) total number of nodes and (d) averaged block size for case 6.
Figure 16: The average block size of cases 1-6 at the performance peak for various machines.
Figure 17: Parallel exaction results: speedup curves of traditional finite element procedure vs. presented cache-efficient ones (CPU time of traditional method was used as reference CPU time for calculation of speedup).

References

  • [1] Ahmed, N., Mateev, N., and Pingali, K. 2000. Synthesizing transformations for locality enhancement of imperfectlynested loop nests. In Conference Proceedings of the 2000 International Conference on Supercomputing, Santa Fe, NM, ACM SIGARCH, pp. 141-152.
  • [2] Alan, G., and Liu, J. 1981. Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall.
  • [3] Bassetti, F., Davis, K., and Quinlan, D. December 1998. Optimizing transformations of stencil operations for parallel object-oriented scientific frameworks on cache-based architectures. In Proceedings of ISCOPE’98, Santa Fe, NM.
  • [4] Bassetti, F., Davis, K., and Marathe, M. December 1999. Improving cache utilization of linear relaxation methods: theory and practice. In Proceedings of ISCOPE’99, San Francisco, CA.
  • [5] Carr, S. and Kennedy, K. 1992. Compiler blockability of numerical algorithms. In Proceedings of 1992 Conference on Supercomputing, Minneapolis, MN, IEEE Computer Society Press, Los Alamitos, CA, pp. 114-124.
  • [6] Comini, G., Guidice, S. D., Lewis, R. W., Zienkiewicz, O. C. 1974. Finite element solution of non-linear heat conduction problems with special reference to phase change. Int. J. Numer. Meth. Eng. 8:613-24.
  • [7] Ding, C. and Kennedy, K. 1999. Improving cache performance in dynamic applications through data and computation reorganization at run time. In Proceedings of the ACMSIGPLAN 1999 Conference on Programming Language Design and Implementation, Atlanta, GA, ACM SIGPLAN Notices, pp. 229-241.
  • [8] Douglas, C. C., Hu, J., Kowarschik, M., Rüde, U., and Weiss, C. 2000a. Cache optimization for structured and unstructured grid multigrid. Electronic Transaction on Numerical Analysis 10:21-40.
  • [9] Douglas, C. C., Hu, J., Iskandarani, M., Kowarschik M., Rüde, U., and Weiss, C. 2000b. Maximizing cache memory usage for multigrid algorithms. Appeared in in Multiphase Flows and Transport in Porous Media: State of the Art, Z. Chen, R. E. Ewing, and Z.-C. Shi (eds.), Lecture Notes in Physics, Vol. 552, Springer-Verlag, Berlin, pp. 234-247.
  • [10] Gannon, D., Jalby, W., and Gallivan, K. 1988. Strategies for cache and local memory management by global program transformation. Journal of Parallel and Distributed Computing 5(5):587-616.
  • [11] Gullerud, A.S., and Dodds Jr, R.H. 2001. MPI-based implementation of a PCG solver using an EBE architecture and preconditioner for implicit, 3-D finite element analysis.Computers and Structures Vol. 79, 553-575.
  • [12] Han, H. and Tseng, C.-W. 2000. Efficient compiler and runtime support for parallel irregular reductions. Parallel Computing 26:1861-1887.
  • [13] Heras, D. B., Perez, V. B., Dominguez, J. C. C., and Rivera, F. F. 1999. Modeling and improving locality for irregular problems: sparse matrix-vector product on cache memories as a case study. In HPCN Europe, Amsterdam, Tne Netherlands, pp. 201-210.
  • [14] Hu, J. J. 2000. Cache based multigrid on unstructured grids in two and three dimensions. Ph.D. Thesis, University of Kentucky, Lexington, Kentucky.
  • [15] Huy, H., and Argyropoulos, S. A. 1996. Mathematical modelling of solidification and melting: a review, Modelling Simul. Mater. Sci. Eng. 4:371-396.
  • [16] Im, E.-J. and Yelick, K. 1999. Optimizing sparse matrix vector multiplication on SMPs. In 9th SIAM Conference on Parallel Processing for Scientific Computing, San Antonio, TX.
  • [17] Im, E.-J. 2000. Optimizing the performance of sparse matrix-vector multiply. Ph.D. Thesis, University of California, Berkeley.
  • [18] Im, E.-J. and Yelick, K. 2001. Optimizing sparse matrix computations for register reuse in sparsity. In Computational Science- ICCS 2001, San Jose, CA, V. N. Alexandrov et al., editors, Lecture Notes in Computer Science, Springer-Verlag, Berlin, pp. 127-136.
  • [19] Im, E. J., Yelick, K., Vuduc, R. 2004. SPARSITY: optimization framework for sparse matrix kernels. International Journal of High Performance Computing Applications 18:135-158.
  • [20] Irigoin, F. and Triolet, R. 1988. Supernode partitioning. In Proceedings of the 15th Annual ACM SIGPLAN Symposium on Principles of Programming Languages, San Diego, CA, pp. 319-329.
  • [21] Karypis, G. and Kumar, V. 1998. Multilevel k-way partitioning scheme for irregular graphs. Journal of Parallel and Distributed Computing 48:96-129
  • [22] Karypis, G., Kumar, V. 2005. METIS: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices version 4.0. http://www.cs.umn.edu/˜karypis [2005].
  • [23] Kodukula, I., Ahmed, N., and Pingali, K. 1997. Data-centric multi-level blocking. In Proceedings of the 1997 ACMSIGPLAN Conference on Programming Language Design and Implementation (PLDI), Las Vegas, NV, pp. 346-357.
  • [24] Leiserson, C. E., Rao, S., and Toledo, S. 1997. Efficient out-ofcore algorithms for linear relaxation using blocking covers. Journal of Computer and System Science 54(2):332-344.
  • [25] Lemmon E. C. 1981. Multidimensional integral phase change approximations for Þnite element conduction codes. Numerical Methods in Heat Transfer, Lewis R W, Morgan K, Zienkiewicz O C (eds). Wiley, New York, pp. 201-213.
  • [26] Lewis, R. W., Morgan, K., Thomas, H. R., and K. N. Seetharamu, K. N. 1996. The finite element method in heat transfer analysis, Wiley, New York.
  • [27] Lewis, R. W., and Ravindran, K. 2000. Finite element simulation of metal casting. Int. J. Numer. Meth. Eng. 47:29-59.
  • [28] Lohner, R. 2001, Applied CFD Techniques: An Introduction Based on Finite Element Methods, John Wiley and Sons, New York.
  • [29] Manzari, M. T., Gethin, D. T., and Lewis, R. W. 2000. Optimization of Heat Transfer Between Casting and Mould, Int. Journal of cast metals Research 13:199-207.
  • [30] McKinley, K. S., Carr, S., and Tseng, C.-W. 1996. Improving data locality with loop transformations. ACM Transactions on Programming Languages and Systems 18(4):424-453.
  • [31] Mellor-Crummey, J., Whalley, D., and Kennedy, K. 2001. Improving memory hierarchy performance for irregular applications using data and computation reorderings. International Journal of Parallel Programming 29:217-247.
  • [32] Mitchell, N., Carter, L., and Ferrante, J. 1999. Localizing nonaffine array references. In Proceedings of the 1999 International Conference on Parallel Architectures and Compilation Techniques, Newport Beach, CA, pp. 192-202.
  • [33] Oliker, L., Li, X., Heber, G., and Biswas, R. 2000. Ordering unstructured meshes for sparse matrix computations on leading parallel systems. In Parallel and Distributed Processing, 15 IPDPS 2000 Workshops, Lecture Notes in Computer Science Vol. 1800, Springer-Verlag, Berlin, pp. 497-503.
  • [34] Pinar, A. and Heath, M. 1999. Improving performance of sparse matrix-vector multiplication. In Proceedings of Supercomputing Portland, OR.
  • [35] Povitsky, A. October 1999 Wavefront cache-friendly algorithm. for compact numerical schemes, Technical Report 99-40, ICASE, Hampton, VA.
  • [36] Pugh, W. and Rosser, E. 1999. Iteration space slicing for locality. In Proceedings of the 12th Workshop on Languages and Compilers for Parallel Computing (LCPC), San Diego, CA.
  • [37] Reddy, J. N., and Gartling, D. K. 2000. The finite element method in heat transfer and fluid dynamics. 2nd edition, CRC Press, London.
  • [38] Rivera, G. and Tseng, C.-W. 2000. Tiling optimizations for 3D scientific computations. In Proceedings of 2000 Conference on Supercomputing, Dallas, TX, ACM/IEEE, New York, pp. 60-61.
  • [39] Sellappa, S., and Chatterjee, S. 2004. Cache-efficient multigrid algorithms. International Journal of High Performance Computing Applications 18:115-133.
  • [40] Song, Y. and Li, Z. 1999. New tiling techniques to improve cache temporal locality. In Proceedings of the ACM SIGPLAN ’99 Conference on Programming Language Design and Implementation, Atlanta, GA, pp. 215-228.
  • [41] Stals, L. and Rüde, U. October 1997. Techniques for improving the data locality of iterative methods, Technical Report MRR 038-97, Institut für Mathematik, Universität Augsburg, Augsburg, Germany.
  • [42] Strout, M. M., Carter, L., Jeanne Ferrante, J., and Kreaseck, B. 2004. Sparse tiling for stationary iterative methods. International Journal of High Performance Computing Applications 18:95-113.
  • [43] Swaminathan, C. R., and Voller, V. R. 1992. A general enthalpy method for modeling solidification processes. Metall. Trans. 23B:651-64.
  • [44] Thomas, B. G., Samarasekara, I. V.,and Brimacombe, J. K. 1984. Comparition of numerical modelling techniques for complex, two-dimensional, transient heat-conduction problems. Metall. Trans. 15B:307-318.
  • [45] Wolf, M. E. and Lam, M. S. 1991. A data locality optimizing algorithm. In Proceedings of ACM SIGPLAN 1991 Conference on Programming Language Design and Implementation, Toronto, Canada, ACM SIGPLAN Notices, Vol. 26, pp. 30-44.
  • [46] Wolfe, M. J. 1987. Iteration space tiling for memory hierarchies. In Proceedings of the 3rd Conference on Parallel Processing for Scientific Computing, Los Angeles, CA, SIAM, Philadelphia, PA, pp. 357-361.
  • [47] Wolfe, M. 1989. More iteration space tiling. In Proceedings of 1989 Conference on Supercomputing, Reno, NV, ACM, New York, pp. 655-664.
  • [48] Wonnacott, D. 2002. Achieving scalable locality with time skewing. International Journal of Parallel Programming 30(3):181-221.
  • [49] Zienkiewicz, O. C., and Taylor, R. L. 2000. The finite element method volume 1: the basis. Fifth edition, Butterworth-Heinemann, Oxford.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
49240
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description