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Supporting Finite Element Analysis with a Relational Database Backend Part II: Database Design and Access Gerd Heber Cornell Theory Center, 638 Rhodes Hall, Cornell University, Ithaca, NY 14853, USA  HYPERLINK "mailto:heber@tc.cornell.edu" heber@tc.cornell.edu Jim Gray Microsoft Research, San Francisco, CA 94105, USA Gray@microsoft.com March 2006 Technical Report MSR-TR-2006-21 Microsoft Research Microsoft Corporation One Microsoft Way Redmond, WA 98052 Supporting Finite Element Analysis with a Relational Database Backend Part II: Database Design and Access Gerd Heber*, Jim Gray *Cornell Theory Center, 638 Rhodes Hall, Ithaca, NY 14853, USA  HYPERLINK "mailto:heber@tc.cornell.edu" heber@tc.cornell.edu Microsoft Research, 455 Market St., San Francisco, CA 94105, USA  HYPERLINK "mailto:Gray@microsoft.com" Gray@microsoft.com Abstract: This is Part II of a three article series on using databases for Finite Element Analysis (FEA). It discusses (1) db design, (2) data loading, (3) typical use cases during grid building, (4) typical use cases during simulation (get and put), (5) typical use cases during analysis (also done in Part III) and some performance measures of these cases. It argues that using a database is simpler to implement than custom data schemas, has better performance because it can use data parallelism, and better supports FEA modularity and tool evolution because database schema evolution, data independence, and self-defining data. Introduction Computational materials science simulations are data-intensive tasks. They start by designing one or more geometric models of parts or structures and generating the simulation finite element mesh(es) or other discretizations. Then the simulation is run, and finally one analyzes the results. A large model can take days to design, weeks or months to run, and months to analyze. Each of these steps may have many subtasks and some of them might be performed in loops. Each requires careful bookkeeping, and each consumes and produces large and complex datasets. Traditionally, materials science simulations, and indeed most scientific tasks have been done using files as the basic data interchange metaphor. Each step of the workflow consumes and produces files uninterpreted streams of bytes. Consequently, analysis tools have large portions of non problem-oriented code that do nothing more than parsing and reformatting byte-streams. We believe basing workflows on uninterpreted byte streams (files) is the wrong approach. Rather, one wants strongly-typed self-defining semantic information stores at each workflow step and one wants a tool that makes it easy to write such strongly-typed self-defining data streams. Indeed, the program steps inputs and outputs are often not streams at all; rather they are spatial, temporal, or conceptual data subsets. So, scientific simulation workflows need to be able to read and write arbitrary subsets or dynamic aggregations of the data store. Databases provide exactly these capabilities. They store strongly-typed (explicitly modeled!) data, they allow arbitrary subsets to be inserted, and they manage very large data stores. They allow multiple organizations and allow easy reorganization. Databases also have other useful features like security, schema evolution (the ability to change data design), protection from hardware and other failures, concurrency control, data scrubbing tools, data mining tools, and non-procedural query tools. We are not saying that files are going away after all, databases are typically stored in files. Rather we are saying that scientists in general, and FEA practitioners in particular, should be working at a higher level of abstraction, they should be able to think in terms of object collections that can be accessed by spatial, temporal, or other attributes. Again, this is an evolution of the concept of a byte stream to the more general notion of an object store. We begin this report with a brief recap of Part I of this three-part series. Subsequently, we introduce three computational geometry problems frequently encountered in FEA and show how to solve the most common one using a modern relational database, SQL Server 2005. This is followed by a performance discussion and some practical considerations. The FEA Workflow and Database The major steps in the FEA workflow include (see [ REF _Ref128197606 \h 19]): Model/geometry generation (CAD etc.) Meshing (= decomposition into simple shapes and voxels) Attribute definition and assignment (material properties, boundary conditions) Equation formulation Solution running the model to produce simulated behavior over time Post-processing (e.g., visualization, feature detection, life prediction) As explained in [ REF _Ref124912285 \h 3], the unstructured finite element analysis (FEA) mesh is part of the simulations metadata. Although the mesh itself makes up the bulk of the metadata, it is typically only less than 5% of the total data (which include simulation results); yet, managing this relatively small subset is crucial for post-processing, analysis, and visualization. It is essential to have a flexible format for it, essential to be able to evolve it, and essential to be able to access it from many different views. There is no standardized representation for finite element meshes: industrial FEA packages use proprietary file formats and end-users spend a good deal of time writing conversion routines and keeping their converters up-to-date with format changes. These file formats are typically optimized for storage space-efficiency and are very different from the layout of in-core structures, which are optimized for fast lookup and access. Designing a database schema for a mesh is much like creating the easy-to-access in-memory structures, with the exception that one does not think in terms of pointers, arrays, or objects, but rather thinks in terms of relations, attributes, and indices. In other words, file formats are not a good guide for schema design. Figure 1 shows the core tables dealing with tetrahedral elements; for simplicity it focuses on the volume mesh relations (it ignores the surface mesh). As mentioned in Part I [ REF _Ref124912285 \h 3], there is more to an FEA model than just the mesh. There is typically some form of parametric geometry or a polyhedral geometric structure; for example, the polycrystal models described in Part III [ REF _Ref124912308 \h 4]. Figure 1: The core tables for a tetrahedral mesh. The key symbols in front of a column name indicate a PRIMARY KEY constraint. The connectors between tables indicate FOREIGN KEY constraints. The Vertices table holds the corners of elements. The Tetrahedra table stores attributes of tetrahedra such as their spatial region identifier (RegionID) and quality measures (volume, JSM). In addition, clues for fast spatial search like the center of mass (x, y, z) and Hilbert code (Hcode) are stored (and indexed). The tetrahedron-vertex relation is stored in the TetrahedronVertices table. Rank is the local ID of a vertex (between 0 and 3 for a tet). If other element types are present (e.g., bricks, wedges, pyramids), similar tables exist for the corresponding element attributes and element-vertex relations. In Part I, the section Mapping an FEA Data Model onto a Relational Model gave a brief discussion of a table layout for the tetrahedron-vertex adjacency relation (a de-normalized quadruple representation and its normalization.) SQL Server 2000 had indirect ways to switch between the normal and quadruple representations using a table-valued function or a triple join. As others [ REF _Ref124912419 \h 10] have pointed out, this translation can be easily expressed as a pivot operation. Using the PIVOT operator now supported in SQL 2005, the (de-normalized) quadruple representation of the tetrahedron-vertex relation can now be easily written as: CREATE VIEW TetQuadRep AS SELECT ElemID, [0] AS v0, [1] AS v1, [2] AS v2, [3] AS v3 FROM TetrahedronVertices PIVOT (SUM(VertexID) FOR Rank IN ([0], [1], [2], [3])) AS PVT Note that the SUM aggregate is completely artificial, since no real aggregation takes place. We could have used MAX or MIN as aggregates with the same result. This query, of course, performs MUCH better than a triple join described in Part I, and there is no longer a need to store both representations, as was necessary for large meshes in SQL Server 2000. Although the UNPIVOT operator can be used to switch from the quadruple representation to the normalized representation, the tetrahedron-vertex representation of Figure 1 should be the one stored, since the quadruple representation cannot be effectively indexed to allow for a fast tetrahedron lookup given a vertex. Getting Data into and out of the Database Given that you have a database schema defined, it is fairly easy to get data into or out of an SQL database. If you have an ascii file named C:\F.csv containing fields separated by tab characters and records delimited by newline characters then the bulk copy command: bcp DB.dbo.T in C:\F.csv c t \t r \n -T imports the data from file F to table T of database DB, and if file Fs data is binary then: bcp DB.dbo.T in C:\F.dat N -T works faster by avoiding the ascii to binary conversion costs (the c means character the N means native and the T means trusted, use my credentials rather than requiring a password). Replacing in by out in the above commands exports the data in table T to file F. Within SQL one can issue a BULK INSERT command that works much like bcp but is inside the SQL process and so is more efficient. Database table sizes are comparable to the binary file representation. In particular, a million row table of ten floating point columns occupies 80 MB in a binary file, 91 MB as an SQL table, and 192 MB in an ASCII file. Of course indices can increase the size of the SQL data, but those indices save IO. Data loading is typically cpu bound. Our system can SQL insert-select a ten-column SQL table into another table at about 20 MB/s (250K records/second). Importing data from a binary file runs at half that speed (about 10 MB/s or 120 k records/s.) Importing data from an ascii file uses nearly twice the cpu to do the conversion and so loads data at about 5MB/s (about 60k records/second). Running multiple import steps in parallel makes sense if you have two or more processors since it is cpu bound, it gets nearly linear speedup. The bcp tool and the bulk insert method do the trick for testing and development as well as the occasional bulk load. But, they are inadequate for building highly automated and robust integrated solutions. The actual workflow of Finite Element Analysis has the following data loading requirements: There are many files in many formats. The source files need to undergo certain transformations before they can be loaded. There is a precedence order in which the some files must be loaded and there are opportunities for concurrent loads (parallelism!). The source files may contain various errors and we need a reliable way to recover. Certain intermediate tasks need to be executed between loads. We dont want human file system monitors or event handlers. This import-export process is repeated again and again as the model develops. Depending on the concurrency and error handling support of your favorite scripting language, just picture how much time and code youll end up writing to produce a robust and fast (parallelism!) solution. These requirements are not unique to FEA, they re-appear in any computational science or analysis application, and they appear in any data warehouse application. So, as you might expect, database systems include a sophisticated tool suite to deal with the Extract-Transform-Load (ETL) problem. SQL Server has a component called Integration Services (SSIS) [ REF _Ref127777170 \h 11, REF _Ref127777172 \h 16] that provides a graphical scripting, debugging, and monitoring system to design and execute ETL tasks. Figure 2 shows a typical SSIS package to move file data into the database. Figure 2: A sample SSIS flow graph. Each icon represents an SSIS task and the connections represent dependences. A task will not start unless all preceding tasks have completed successfully (failure dependencies are supported but not used in this package.) Independent tasks run in parallel. Most of the tasks shown are Bulk Insert tasks. There are also two Execute Process and two Execute SQL tasks. The first Execute Process task, VLFRMeshMaps2csv, runs a Python script which splits and reformats the mesh generator output. All Bulk Insert tasks depend upon its successful completion. The second Execute Process task, CSV cleanup, deletes intermediate processing files. It will run only if all Bulk Insert tasks have successfully completed. The two Execute SQL tasks execute stored procedures which initialize the mesh edges and tetrahedra. The SSIS package can be stored in a database or in a file. SSIS packages are typically executed from the command-line, the GUI, or a scheduled job under the control of the SQL Server Agent. SSIS packages are fully customizable through user-defined variables and transformations. For example, the package shown has 6 user-defined variables which parameterize the destination database server and database as well as the source directories for data, format files, python scripts and a filemask. Getting the Input Data to the Simulation Larger FEA typically execute as an MPI job on a Beowulf computer cluster. Traditionally, on simulation start, each node of the cluster reads its partition of the mesh from a specific file or from a specific offset within a file. The number of partitions may be less, equal, or greater than the number of MPI processes. The initial partitioning might have been obtained by a cheap method like recursive coordinate bisection [ REF _Ref126145642 \h 13] or it might be a left-over from a previous run, and the final partitioning is the result of a high-quality in-memory partitioner like ParMETIS [ REF _Ref126377544 \h 14]. Partitioning is easier and more flexible when the mesh is stored in a database indexed by some spatial parameter. We use a tetrahedrons center of mass (a point) on a Hilbert space-filling curve [ REF _Ref124912435 \h 5] as its spatial key. Each tetrahedrons Hilbert code becomes the partitioning key. In this way it is easy to create any number of spatial partitions using the SQL Server 2005 row ranking functions (ROW_NUMBER, RANK, DENSE_RANK, NTILE). We use the NTILE as a simple partitioning routine. For example the query: SELECT ElemID, NTILE(256) OVER (ORDER BY Hcode) AS partition FROM Tetrahedra returns pairs of element ID and partition ID (between 1 and 256) based on the rank of a tet on the Hilbert curve. Partitioners based on space-filling curves are standard in parallel toolkits like Zoltan [ REF _Ref126145642 \h 13] and for most practical purposes the initial partitioning obtained this way has adequate balance. The primary goal here is to get a balanced startup scenario and nothing prevents us from adaptively re-partitioning in memory (with ParMETIS). Partitioning is a good example of using the database to simplify the application design, execution and management. As explained before, many simulations spend considerable code and execution time on file handling. The use of a database system has the dual benefit of eliminating a lot of code and also giving automatic parallelism to reduce execution time. Long-running simulations periodically create output (load steps or time steps) and check point/restart information. The compute nodes local storage is insufficient to absorb the result sets from several days or weeks of processing which is why the results are written to dedicated I/O nodes or large scratch volumes. SQL Server Integration Services packages periodically wake up and copy recent simulation output data to the database where it becomes part of the model output and can be analyzed while the simulation progresses. How Databases Help Solve Some Standard Computational Geometry Problems in FEA Typical computational geometry problems in FEA deal with locating points inside elements and interpolating field values at (those) arbitrary points. Below, we formulate three typical FEA computational geometry problems and describe the implementation in SQL Server 2005 of Problem 1 in detail. We find that a set-oriented non-procedural language saves writing a lot of code, and overall executes much faster than a file-oriented approach. To be precise, the problem is not so much that traditional approaches store the data in files. The issues are the following: In order to solve the problems described below, search and other auxiliary structures must be created in core. They must be created each time from scratch or they are persisted in some form adding more complexity. Building these indices takes programming time and execution time in the worst case, every time a batch of objects is processed this execution overhead is incurred. On a 32-bit architecture one quickly runs out of memory. Writing efficient out-of-core file processing code is not for the faint-hearted. Even on a 64-bit architecture, the sizes of datasets are often larger than the affordable RAM. Parallelism can substantially increase throughput. Implementing such parallelism in a file-oriented design increases coding complexity. Database systems create search structures (mostly indexes) as part of the initial setup. These structures work well even if most of the index and data are not memory-resident. SQL automatically allows parallelism with no extra programming if it is used as a set-oriented language. Unless record-at-a-time cursors are used in a SQL query, the optimizer automatically detects and exploits processor and disk parallelism. Now lets look at some specific examples of using the database to simplify FEA programming. Problem 1: The Point-in-Tetrahedron Query for Unstructured Tetrahedral Meshes Finding the mesh tetrahedron containing a given point is central to most FEA post-processing, especially visualization. Papadomanolakis et al. propose a novel solution to the point-in-cell problem for unstructured tetrahedral meshes called Directed Local Search (DLS) [ REF _Ref124912323 \h 7]. DLS bears little resemblance to traditional B-, R-, P-tree methods. It first makes a well-informed choice of a candidate tetrahedron and then uses a tetrahedron-connectivity-graph traversal technique to quickly find the containing tetrahedron. DLS is a hybrid algorithm exploiting both geometric and connectivity information. Papadomanolakis et al. implemented DLS in PostgreSQL [ REF _Ref124912353 \h 8] and compared it with other methods [ REF _Ref124912323 \h 7]. A detailed description of our DLS implementation in SQL Server 2005 is provided below. Problem 2: Interpolation of a Field Value at an Arbitrary Point from an Unstructured Mesh FEA approximates complex geometries and continuous fields by assigning values to isolated points in space (nodes). It uses interpolation to approximate the fields value at any point inside the mesh. In FEA terminology, the interpolation scheme is called a shape function. Roughly speaking, the value  EMBED Equation.DSMT4  of a field at a point  EMBED Equation.DSMT4  inside an element  EMBED Equation.DSMT4  is given as the weighted sum of the values  EMBED Equation.DSMT4  attached to the nodes  EMBED Equation.DSMT4  of the element, like element corners and midpoints of edges.  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 1) The weights  EMBED Equation.DSMT4  are the values of the elements shape functions  EMBED Equation.DSMT4  at the point EMBED Equation.DSMT4 . In particular, the shape functions are not constant across an element and, in general, are non-linear functions of a points coordinates. Field interpolation is an essential part of post-processing and visualization. Given an arbitrary point  EMBED Equation.DSMT4 , the point-in-tetrahedron query identifies the mesh element containing  EMBED Equation.DSMT4 . One can then compute the points field values using equation (1) and the nodes shape functions at  EMBED Equation.DSMT4 . If the shape functions are non-linear, a (small) system of non-linear equations needs to be solved to determine their values at  EMBED Equation.DSMT4 . Problem 3: Interpolation of a Value from a Cloud of Points Reference [12], formulates the value-from-a-cloud-of-points problem as follows: Given a set of randomly distributed points  EMBED Equation.DSMT4  with associated scalar values  EMBED Equation.DSMT4  interpolate a value  EMBED Equation.DSMT4  at point EMBED Equation.DSMT4  within the convex hull of EMBED Equation.DSMT4 . (The convex hull is the smallest convex set which contains the point set.) This problem naturally arises when experimental data must be interpolated onto a computer model. The model time-space geometry might be simplified and there is no one-to-one match between the measurement points and the model grid points. Though formally very similar to the interpolation problem for unstructured meshes (Problem 2), the point cloud turns out to be more difficult. We compensate for the lack of pre-defined interpolation paths (connectivity) by introducing a spatial index (octree) to quickly find nearby points. The number of interpolation points depends on the desired interpolation order: Four points are adequate for or linear interpolation. More points are required for higher order interpolation. Unfortunately, proximity alone is not sufficient: certain dependences among the closest points like colinearity or complanarity will lead to singular covariance matrices in which case alternative points are required [12]. The Point-in-Cell Query for Unstructured Tetrahedral Meshes using Directed Local Search (DLS) The idea behind DLS can be stated as: let  EMBED Equation.DSMT4  be a point with Cartesian coordinates EMBED Equation.DSMT4 . Assume that we know a priori that there exists at least one mesh element that contains p. If the volume cannot be easily approximated by piecewise low-order or polygonal geometries, this decision is costly; but for our polycrystal models (see [ REF _Ref124912308 \h 4]) the answer is a trivial point-in-box test. The DLS method finds a tetrahedron containing  EMBED Equation.DSMT4  in two steps: Find a good candidate tetrahedron  EMBED Equation.DSMT4  in the vicinity of EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4  turns out not to contain  EMBED Equation.DSMT4 , use the tetrahedron-face connection graph to pick an adjacent tetrahedron  EMBED Equation.DSMT4  in the direction defined by the vector from the center of mass  EMBED Equation.DSMT4  of the current tetrahedron to the point  EMBED Equation.DSMT4  and repeat step 2 using  EMBED Equation.DSMT4  as  EMBED Equation.DSMT4  until the containing tetrahedron is found. Candidate Selection A common technique to find (small) spatial objects near a point  EMBED Equation.DSMT4  is to compare their ranks on a Hilbert space-filling curve [ REF _Ref124912435 \h 5]. An (approximate) Hilbert curve can be used to define a ranking or linearization  EMBED Equation.DSMT4  of points  EMBED Equation.DSMT4  of a (non-negative) cubic lattice. The locality preserving nature of  EMBED Equation.DSMT4  allows us to be optimistic: if the difference between the ranks  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  of two points is small, then their Euclidean distance between points  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  is often small. This makes  EMBED Equation.DSMT4  an excellent choice for a candidate selector. For each tetrahedron we pre-compute, store, and index (!) the  EMBED Equation.DSMT4  code of its center of mass. If a point is close to the center of mass of a tetrahedron, then there is a fair chance (the bigger the tet the better!) that it falls inside the tetrahedron. Note that neither a  EMBED Equation.DSMT4 -match nor an  EMBED Equation.DSMT4 -difference of a point and a tetrahedrons centroid yields any conclusion about the inclusion of the point in the tetrahedron. However, a simple algebraic test can establish that: Let  EMBED Equation.DSMT4  be the corners of a tetrahedron and  EMBED Equation.DSMT4  (vectors). If the tetrahedron is non-degenerate (= has non-zero volume), an arbitrary point  EMBED Equation.DSMT4  can be uniquely written as  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 2) (This is a system of three linear equations for three unknowns EMBED Equation.DSMT4 .)  EMBED Equation.DSMT4  is inside the tetrahedron defined by  EMBED Equation.DSMT4 , iff  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Solving for , ,  involves ~71 floating point computations (four  EMBED Equation.DSMT4 determinants and three divisions) which are very fast on modern processors. Traversal If the candidate tetrahedron, EMBED Equation.DSMT4 , contains the point  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 is the answer to the tet-contains-point query. If the containment test fails for  EMBED Equation.DSMT4 , the line from the centroid of  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4  intersects at least one of the tetrahedrons faces. Make the adjacent tetrahedron the new candidate. To implement this, let  EMBED Equation.DSMT4  be the centroid of  EMBED Equation.DSMT4 . The ray defined by direction (vector)  EMBED Equation.DSMT4  intersects the triangular facet EMBED Equation.DSMT4 , iff  EMBED Equation.DSMT4 can be represented as  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 3) We examine each of the four faces of  EMBED Equation.DSMT4  until we find a solution to this equation. Unless the triangle we find is on the outer surface of the body, there is exactly one tetrahedron on the other side of the intersected face. We make this neighbor our current tetrahedron and repeat the inclusion test. Figure 3 illustrates the traversal technique in 2D. Figure 3: A sample 2D traversal. The shaded triangle represents the candidate triangle. It does not contain  EMBED Equation.DSMT4 . The line from the center of gravity of the candidate to  EMBED Equation.DSMT4  intersects the edge  EMBED Equation.DSMT4 . The triangle on the other side of  EMBED Equation.DSMT4 does not contain  EMBED Equation.DSMT4  either and we have to continue iterating until we reach the triangle that contains  EMBED Equation.DSMT4 . EMBED Visio.Drawing.11 A DLS Implementation in SQL Server 2005 Based on the description in the previous section, the following routines need to be implemented: A function to calculate the Hilbert code,  EMBED Equation.DSMT4 , of a point  EMBED Equation.DSMT4  A Boolean function that implements the point-in-tet test A routine that, given a ray r and a tetrahedron t, determines which facet(s) of t are intersected by r. A routine which determines the ID of the tetrahedron on the other side of a tetrahedron face Calculating the  EMBED Equation.DSMT4  code Reference [ REF _Ref124912435 \h 5] describes an elegant implementation of  EMBED Equation.DSMT4 . We converted it to C# (about 500 lines) and made some minor adaptations: In the C implementation of [ REF _Ref124912435 \h 5], the  EMBED Equation.DSMT4  code is represented as a structure of three unsigned 32-bit integers. The largest scalar integral type in SQL is a signed 64-bit bigint. We condensed three unsigned 20-bit integers into a bigint  EMBED Equation.DSMT4  code. This gives us a resolution of  EMBED Equation.DSMT4  grid points in each coordinate direction. The C# prototype looks as follows: public class Hilbert3D { // we use only the lower 20 bits of x, y, z public static UInt64 H_encode(UInt32 x, UInt32 y, UInt32 z); // this is what we will call from T-SQL [SqlFunction(IsDeterministic = true, IsPrecise = true, DataAccess = DataAccessKind.None)] public static SqlInt64 H_encode_SQL(SqlInt32 x, SqlInt32 y, SqlInt32 z) { return H_encode((UInt32)x, (UInt32)y, (UInt32)z)); } } This defines the H_encodeSQL function signature and tells SQL that the function is deterministic and precise, so that it can optimize the function call and execute it in parallel. After registering the DLL with SQL Server, the encoding function can be made accessible to T-SQL via: CREATE FUNCTION fnH_encode(@x int, @y int, @z int) RETURNS bigint AS EXTERNAL NAME [SFC].[SFC.Hilbert3D].[H_encode_SQL] (The DLLs name is SFC.dll and Hilbert3D is a class in a namespace called SFC.) We calculate  EMBED Equation.DSMT4  for each tetrahedron and create a clustered index on the Hcode column: CREATE TABLE Tetrahedra ( ElemID int PRIMARY NONCLUSTERED KEY, X float NOT NULL, y float NOT NULL, z float NOT NULL, Hcode bigint NOT NULL, ...) UPDATE Tetrahedra SET Hcode = dbo.fnH_encode(x, y, z) CREATE CLUSTERED INDEX Idx_Hcode ON Tetrahedra(Hcode) The  EMBED Equation.DSMT4  calculation takes about  EMBED Equation.DSMT4  per row. An update takes about  EMBED Equation.DSMT4  per row. As pointed out in [ REF _Ref124912323 \h 7], a partial index would be sufficient. For example, randomly selecting an adjacent tetrahedron for each vertex in the mesh would provide an adequate subset to index and to make DLS work. Point-in-Tet Test For the point-in-tet test we solve equation  GOTOBUTTON ZEqnNum428034 \* MERGEFORMAT  REF ZEqnNum428034 \! \* MERGEFORMAT (2) and check whether  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . The test returns a T-SQL Boolean value indicating the result. Note that the test also returns 1 also if the point falls on the tetrahedrons face, edge, or vertex. To accomplish that (in particular for high aspect ratio tets!), the inequalities for  EMBED Equation.DSMT4  are augmented with a tolerance EMBED Equation.DSMT4 . A value of  EMBED Equation.DSMT4  worked well for us in practise. The C# prototype of this T-SQL user defined function is: CREATE FUNCTION fnPointInTet(@ElemID int, @x float, @y float, @z float) RETURNS bit AS EXTERNAL NAME FEMUtils.FEMUtils.Tetrahedron.PointInTet reads as follows: [SqlFunction(IsDeterministic = true, IsPrecise = true, DataAccess = DataAccessKind.Read)] public static SqlBoolean PointInTet(SqlInt32 ElemID, SqlDouble x, SqlDouble y, SqlDouble z); Note that the data access attribute is now set to Read since inside the method we have to fetch the corner coordinates in order to formulate equation  GOTOBUTTON ZEqnNum428034 \* MERGEFORMAT  REF ZEqnNum428034 \! \* MERGEFORMAT (2). Since the code is executing on the server in the context of the user-defined funtion, there is no need to create a special loopback connection. We bind to this context connection as follows: using (SqlConnection conn = new SqlConnection("context connection=true;")) The RayIntersectsTetFace Test We determine which face is intersected by solving equation  GOTOBUTTON ZEqnNum658331 \* MERGEFORMAT  REF ZEqnNum658331 \! \* MERGEFORMAT (3) for each triangular face until we find one where  EMBED Equation.DSMT4 . The test returns the local face ID (0,1,2,3) of the intersected face or returns 4, if the point is actually inside the tetrahedron: CREATE FUNCTION fnRayIntersectsTetFace(@ElemID int, @x float, @y float, @z float) RETURNS tinyint AS EXTERNAL NAME FEMUtils.FEMUtils.Tetrahedron.RayIntersectsTetFace The C# prototype is virtually identical to the point-in-tet test, except that the return value is of type SqlByte to represent a tinyint in T-SQL. [SqlFunction(IsDeterministic = true, IsPrecise = true, DataAccess = DataAccessKind.Read)] public static SqlByte RayIntersectsTetFace(SqlInt32 ElemID, SqlDouble x, SqlDouble y, SqlDouble z); The GetTetFaceNeighbor Lookup Papadomanolakis [ REF _Ref124912323 \h 7] suggests storing the IDs of each tetrahedrons face-neighbors. We do not believe that this is practical: mesh generators typically do not generate this information and calculating it from scratch for large meshes is expensive especially since little of the information would ever be used. A lazy evaluation strategy, one where we compute the neighbors as needed and save them for future lookup, seems more reasonable. We wrote a T-SQL function fnGetTetFaceNeighbor which, given an element ID and the rank of a face, determines the ID of the tetrahedron on the other side or returns -1 in case the face is a triangle on the surface. At its heart is the TetrahedronVertices relation: CREATE TABLE TetrahedronVertices ( ElemID int NOT NULL REFERENCES Tetrahedra, Rank tinyint NOT NULL CHECK(Rank BETWEEN 0 AND 3), VertexID int NOT NULL REFERENCES Vertices, CONSTRAINT PK_TetrahedronVertices PRIMARY KEY (ElemID, Rank), CONSTRAINT UNQ_TetrahedronVertices UNIQUE (ElemID, VertexID) ) The following index improves the performance of fnGetTetFaceNeighbor by giving a fast adjacent element lookup given a vertex: CREATE INDEX Idx_VertexID ON TetrahedronVertices (VertexID) The  EMBED Equation.DSMT4 -th face of a tetrahedron  EMBED Equation.DSMT4  is the face opposite to the  EMBED Equation.DSMT4 -th corner. Given an element and one of its vertices, we are looking for another element which shares the other three vertices. Unless the face opposite to the vertex is on the outer surface, there is exactly one tetrahedron with that property, and its ID is what the fnGetTetFaceNeighbor T-SQL function below returns: CREATE FUNCTION fnGetTetFaceNeighbor(@ElemID int, @oppositeVertex tinyint) RETURNS int AS BEGIN DECLARE @result int -- result is ID of desired tet SELECT @result = ElemID -- find the first face FROM TetrahedronVertices -- in the Tet-Vertex table WHERE VertexID IN ( -- where all the vertices in common with SELECT VertexID -- one of the vertices of @ElemID FROM TetrahedronVertices -- (this select statment returns WHERE ElemID = @ElemID -- all the vertices of the tet) AND Rank != @ oppositeVertex) -- excluding the one opposite the face AND ElemID != @ElemID -- it is a different face GROUP BY ElemID -- group matching vertices HAVING COUNT(VertexID) = 3 -- insist all 3 verticies match RETURN COALESCE(@result, -1) -- return -1 if no such element END The  EMBED Equation.DSMT4 -th face of a tetrahedron  EMBED Equation.DSMT4  is the face opposite to the  EMBED Equation.DSMT4 -th corner. Given an element and one of its vertices, we are looking for another element which shares the other three vertices. Unless the face opposite to the vertex is on the outer surface, there is exactly one tetrahedron with that property, and its ID is what fnGetTetFaceNeighbor returns. Performance of DLS One of the metrics suggested in [ REF _Ref124912323 \h 7] is page accesses per point query. Since this report is aimed at end-users and we assume that our metadata (the mesh) will often fit in memory, we prefer a more application-oriented metric, namely points per second. Our test setup is as follows: We consider a fixed mesh from one of our polycrystal models (see Figure 4). Figure 4: Sample model and mesh. In the top figure, a polycrystal geometry [ REF _Ref124912308 \h 4] is shown. (Data by courtesy of A.D. Rollett and S. Sintay, Carnegie Mellon University) The individual crystals or grains have been isolated for visualization purposes. The figure below shows the surface mesh. The grains themselves are filled with tetrahedra, abutting the triangles on the surface. The mesh has 8,782,315 tetrahedra and 1,520,308 vertices. (In Appendix B, we describe a procedure how to generate the triangle set on the outer surface of the polycrystal with a simple SQL query.) Because of the rich surface structure of the grains, the triangle (and tetrahedron) density is highly non-uniform. The non-uniformity in object density and the unbounded discrepancy in shape between a coordinate-plane aligned box and a tetrahedron is what makes the point-in-cell test for unstructured tetrahedral meshes so difficult. The mesh shown in Figure 4 has 8,782,315 tetrahedra and 1,520,308 vertices. It is very non-uniform in terms of element size and connectivity. A vertex is shared on average by 23 tetrahedra, with a range from 2 to 192 and a variance of 67. The geometry of this particular grid is a brick with the dimensions:  EMBED Equation.DSMT4 . For each pair of center  EMBED Equation.DSMT4  and radius  EMBED Equation.DSMT4 , we generated a table T(x float, y float, z float) containing a spherical cloud of uniformly distributed random points. That is, the points  EMBED Equation.DSMT4  in the cloud satisfy:  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 4) Each table contained 20,000 points and for each such table T we ran the following query: SELECT COUNT(DISTINCT dbo.fnGetTet4Point(x, y, z)) FROM T (fnGetTet4Point is the user-defined function that wraps all of the above functions and returns the ID of a tetrahedron containing the point  EMBED Equation.DSMT4 .) This query returns the number of distinct tetrahedron IDs encountered when categorizing the points of the point cloud from the sample table. Table 1 shows the elapsed runtime and points per second (execution time divided by the number of points in the table) for different centers and radii. We also recorded the average number of number of distinct tetrahedron IDs returned by the query. Table  SEQ Table \* ARABIC 1: Points per second and distinct tet count for sample clouds. EMBED Equation.DSMT4  EMBED Equation.DSMT4 Points / s#Distinct Tetrahedra(2.1, 1.35, 0.65)0.0000128571(2.1, 1.35, 0.65)0.000122222(2.1, 1.35, 0.65)0.00118182(2.1, 1.35, 0.65)0.017418(2.1, 1.35, 0.65)0.1689294(2.1, 1.35, 0.65)0.256891831(2.1, 1.35, 0.65)0.54762443(2.1, 1.35, 0.65)0.64262889(0, 0, 0)0.54163249The results in Table 1 confirm what one would expect: the smaller the radius and the fewer distinct tetrahedra the higher the throughput (points per second). In that sense, the first result in Table 1 comes close to peak performance where we are dealing effectively with one tetrahedron. We expect the cache hit rate to degrade with increasing radius and an increasing number of distinct tetrahedra in the result set. The exact degradation is controlled by the local geometry, i.e., the number, size, and shape of the tetrahedra near  EMBED Equation.DSMT4  and the random points. To account for the local variability in geometry we generated point clouds of random diameter in random locations. The maximum radius depends on the model scale and the application. For our model scale and the intended application (low order screening) a maximum radius of 0.1 is more than generous. Given a fixed total number of points (20,000), the locality behavior will change depending on whether we have more points clustered in fewer locations or fewer points clustered in more locations. Table 2 shows the results for randomly centered and sized point clouds: we generated  EMBED Equation.DSMT4  uniformly distributed points  EMBED Equation.DSMT4  and radii  EMBED Equation.DSMT4  (with mean  EMBED Equation.DSMT4  and standard deviation  EMBED Equation.DSMT4 ), and  EMBED Equation.DSMT4  points per cloud, where  EMBED Equation.DSMT4 . Table  SEQ Table \* ARABIC 2: Points per second and distinct tet count for random sample clouds EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 Points / s#Distinct Tetrahedra1020000.03120.0130833542010000.02990.01425881592001000.02670.014825013982000100.02460.014520843891020000.06230.02615561382010000.05980.02845714122001000.05340.029631224122000100.04910.02891675405200001N/AN/A757392The Hcode (clustered) index was very selective in these experiments: no two tetrahedra had the same Hcode. Practical Considerations Officially only assemblies from source code written in C# and/or VB.NET are supported in SQL Server 2005. Our attempts to deploy assemblies generated with /clr:safe from C++/CLI [ REF _Ref127784392 \h 17] codes failed repeatedly. For scientific and engineering applications, this is a severe limitation: there is a substantial C++ code base and the conversion from C++ templates to C# generics is tedious and, in some cases, impossible. When creating an assembly in SQL Server, access security permissions must be specified. The options are SAFE, EXTERNAL ACCESS, and UNSAFE. SAFE is the default. All the examples in this report are SAFE, no access to the file system, the network, or unmanaged code was allowed or requried. EXTERNAL ACCESS relaxes this restriction and allows access, for example, to the file system through managed and verifiable .NET interfaces. Finally, the UNSAFE permission set allows calling unmanaged code (e.g. using P/Invoke). This creates several opportunities to integrate data from diverse sources under the umbrella of SQL Server. For example, data can be read from or written to netCDF [ REF _Ref128279283 \h 20] files by calling routines in the (unmanaged) netCDF DLL from table-valued functions or stored procedures. Got XML data? There is no need anymore to work with textual representations of XML documents or shred them to fit a tabular model. It is beyond the scope of this report, but we dont want to close without encouraging you to explore SQL Servers XML type and XQuery support (and many of the other new features!). Summary This discussion of the database aspects of FEA started by explaining that many current programs spend considerable code and time on data parsing and that the evolution of these programs is hampered by need to make bi-lateral changes to the data producer and the data consumer. Using self-defining data schemas and a relational database interface gives FEA codes the same data independence advantages that they give to other applications. These data independence and schema evolution issues are particularly important for the FEA mesh and its associated attributes. Databases tools are also quite useful for data ingest. The next article in this series shows how helpful the database representation is for analysis and visualization tools. But, for the simulation steps, the most important aspects are the data independence, parallel database access (for performance), and the ease with which one can specify fairly complex data access concepts. This article gave several examples of complex data access ranging from the dual representation of tetrahedra using the pivot operator the normalized face representation and the denormalized quad representation. It also showed how directed local search leverages relational operators and indices to quickly implement spatial search. These are not isolated examples, they are chosen as representatives form a large collection of techniques we have used to build the FEA system behind our Computational Materials system COMPASS [ REF _Ref128370844 \h 25]. That system is a considerable advance over its predecessor and demonstrates the benefits of using a database representation for the FEA model, grid, results, and beyond. Acknowledgements We gratefully acknowledge the support of the Cornell Theory Center, Microsoft, and Microsoft Research. The first author would like to acknowledge the support received under the DARPA SIPS program. References  SEQ [ \* ARABIC 1.DeBetta, P.: Introducing Microsoft SQL Server 2005 for Developers. ISBN: 073561962X, Microsoft Press (2005) SEQ [ \* ARABIC 2.HYPERLINK "http://research.microsoft.com/BARC/Sequential_IO/SataDiskIO.doc"Gray, J. and Kukol, P.: Sequential Disk IO Tests for GBps Land Speed record. (May 2004) SEQ [ \* ARABIC 3. HYPERLINK "http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSR-TR-2005-49" Heber, G., Gray, J.: Supporting Finite Element Analysis with a Relational Database Backend Part I. Microsoft Research Technical Report, MSR-TR-2005-49 (2005) SEQ [ \* ARABIC 4. HYPERLINK "http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSR-TR-2005-151" Heber, G., Gray, J.: Supporting Finite Element Analysis with a Relational Database Backend Part III. Microsoft Research Technical Report, MSR-TR-2005-151 (2005) SEQ [ \* ARABIC 5.Lawder, J.K.: Calculation of Mapping Between One and n-dimensional Values Using the Hilbert Space-filling Curve. Research Report BBKCS-00-01 (2000) SEQ [ \* ARABIC 6. HYPERLINK "http://approjects.co.za/?big=sql/default.mspx" Microsoft SQL Server Home SEQ [ \* ARABIC 7.Papadomanolakis E. et al.: Efficient Query Processing on Unstructured Tetrahedral Meshes. To appear SEQ [ \* ARABIC 8. HYPERLINK "http://www.postgresql.org/" PostgreSQL: The worlds most advanced open source database (2006) SEQ [ \* ARABIC 9.Rizzo, T. et al.: Pro SQL Server 2005. ISBN: 1590594770, Apress (2005) SEQ [ \* ARABIC 10.Tropashko, V., Oracle Corporation, Private communication (September 2005) SEQ [ \* ARABIC 11.Farmer, D.: The Rational Guide to Scripting SQL Server 2005 Integration Services. ISBN: 1932577211, Rational Press (2005) SEQ [ \* ARABIC 12. HYPERLINK "http://www.andrew.cmu.edu/user/sowen/abstracts/Ba960.html" Baker, T.J.: Interpolation from a cloud of points. Proceedings 12th International Meshing Roundtable, September 14-17 2003, Santa Fe, New Mexico, pp. 55-63 (2003) SEQ [ \* ARABIC 13. HYPERLINK "http://www.cs.sandia.gov/Zoltan/" Zoltan: Data-Management Services for Parallel Applications. Sandia National Laboratories SEQ [ \* ARABIC 14. HYPERLINK "http://www-users.cs.umn.edu/~karypis/metis/parmetis/index.html" ParMETIS: Parallel Graph Partitioning & Sparse Matrix Ordering SEQ [ \* ARABIC 15.Dewson, R. and Skinner, J.: Pro SQL Server 2005 Assemblies. ISBN: 1-59059-566-1, Apress (2006) SEQ [ \* ARABIC 16.Knight, B. et al.: Professional SQL Server 2005 Integration Services. ISBN: 0-76458-435-9, Wiley (2006) SEQ [ \* ARABIC 17.Fraser, S.R.G.: Pro Visual C++/CLI and the .NET 2.0 Platform. ISBN: 1-59059-640-4, Apress (2006) SEQ [ \* ARABIC 18.Wawrzynek, P.A.: FemLib Finite Element Library. Cornell Fracture Group SEQ [ \* ARABIC 19.Rennet, A. and Zieglar M.: Data Management Systems for FEA Data. BENCHmark Magazine July 2003, pp. 8-12 (2003) SEQ [ \* ARABIC 20. HYPERLINK "http://www.unidata.ucar.edu/software/netcdf/" NetCDF (network Common Data Form) SEQ [ \* ARABIC 21.Van der Waerden, B. L.: Algebra. Springer-Verlag (1991) SEQ [ \* ARABIC 22.Seifert H. and Threlfall, W.: Textbook of Topology. Academic Press (1980) SEQ [ \* ARABIC 23.Hilbert, D. and Cohn-Vossen, S.: Geometry and the Imagination. Chelsea (1952) SEQ [ \* ARABIC 24. HYPERLINK "http://approjects.co.za/?big=downloads/details.aspx?familyid=9a8b005b-84e4-4f24-8d65-cb53442d9e19&displaylang=en" SQLIO Disk Subsystem Benchmark Tool SEQ [ \* ARABIC 25.COMPASS COmputational Materials Portal and Adaptive Simulation System. Cornell Fracture Group (2005) Appendix A: System Configuration The system used for all the tests in this report in its configuration and performance is very similar to the one dubbed Tyan/Opteron/Marvell Configuration in [ REF _Ref127161090 \h 2]. Hardware Tyan Thunder K8SD Pro S2882-D motherboard 2 x Dual Core AMD Opteron 275 (2.2 GHz, 1 MB L2 cache per core) 8 x 1 GB DDR-SDRAM memory 3 x Supermicro 8-port SATA card (AOC-SAT2-MV8) 24 x Hitachi 500GB HDS725050KLA360 hard drives Software Microsoft Windows Server 2003 R2 Enterprise x64 Edition Microsoft SQL Server 2005 Enterprise x64 Edition Microsoft .NET Framework 2.0 (build 2.0.50727) The attached disk storage is configured as JBOD (24 NTFS mount points). We ran SQLIO [ REF _Ref128290397 \h 24] to determine how the aggregate bandwidth scales across the three SATA controllers (see Figure 5).  EMBED Excel.Chart.8 \s Figure 5: SQLIO performance. The bandwidth was determined by reading or writing for 30 seconds 256KB blocks sequentially to files on multiple disks. Both read and write aggregate bandwidth scale almost linearly across all three controllers and 24 disks. The single disk performance for read and write was slightly over 61 MB/s. The top rate across 24 disks was around 1.2 GB/s for read and write. The CPU utilization while simultaneously accessing all 24 disks was around 18% (4 cores = 100%). Appendix B: Recovering the Surface Mesh of a Tetrahedral Mesh A common (non-geometric) problem in FEA is recovering the surface mesh given only a tetrahedral mesh. Each tetrahedron is bounded by four triangular facets and such a facet is considered part of the surface mesh, if and only if it bounds exactly one tetrahedron. (Internal triangular facets are shared by exactly two tetrahedra: a triangular facet shared by less than one or more than two tetrahedra indicates an error in the connectivity.) A standard approach is to create a hash table where the keys are triangles and the values count how many elements share the key triangle. If the connectivity is correct, there are only two possible values (1 and 2) and, after all triangles have been processed, the triangle keys with value 1 are the ones that make up the surface mesh. There are a few subtleties and refinements of the problem that complicate matters somewhat: A triangle is typically represented as a triplet of its corner vertex IDs  EMBED Equation.DSMT4 . Do  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  represent the same triangle? It depends The surface of a volume mesh is typically oriented, i.e., for each triangle on the surface we pick an order of its vertex IDs such that all resulting geometric normals either point into the interior of the (volumetric) domain or into the exterior. Under these assumptions,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  do not represent the same (oriented) triangle, because the normal (vector product)  EMBED Equation.DSMT4  has the opposite sign of  EMBED Equation.DSMT4 . Incoherently oriented surface meshes may cause rendering (shading) artifacts in visualization applications and incorrect results in numerical integration we typically have a keen interest in generating a coherently oriented surface mesh. We solve this problem in two steps: we first generate a set of normalized un-oriented triangles representing the (un-oriented) surface mesh and then, in a second step, recover the orientations from the tetrahedral volume mesh. The triplet representation  EMBED Equation.DSMT4  can be easily normalized by imposing a constraint like  EMBED Equation.DSMT4 . The set of (normalized) triplets corresponding to the triangles in the surface mesh can be easily generated as follows: SELECT A.VertexID AS [0], B.VertexID AS [1], C.VertexID AS [2] -- select the 3 verticies FROM TetrahedronVertices AS A -- by insisting they are part of JOIN TetrahedronVertices AS B ON A.ElemID = B.ElemID -- the same element JOIN TetrahedronVertices AS C ON B.ElemID = C.ElemID -- and WHERE A.VertexID < B.VertexID AND B.VertexID < C.VertexID -- that thery are in ascending order GROUP BY A.VertexID, B.VertexID, C.VertexID -- consider all triplets HAVING COUNT(A.ElemID) = 1 -- that occurr in a single element The optimizer creates a rather efficient execution plan for this query: for our volume mesh of 8,782,315 tetrahedra, 147,708 triangles are generated in 290 seconds at an average CPU utilization above 97% (each CPU core accounts for 25%). If all we want is an un-oriented surface mesh, we are done. To get the orientations right, we have to understand where they come from and why the triplet representation is a little off the mark. A tetrahedron can be represented as a quadruple of its corner vertex IDs  EMBED Equation.DSMT4 . Strictly speaking, an oriented tetrahedron is an equivalence class of such quadruples: two such quadruples are considered equivalent if they differ only by an even permutation of their entries (components). For example, the quadruples  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  represent the same oriented tetrahedron. On the other hand, the quadruple  EMBED Equation.DSMT4  represents a tetrahedron of the opposite orientation (since a transposition is an odd permutation). The oriented boundary  EMBED Equation.DSMT4  of an oriented tetrahedron  EMBED Equation.DSMT4  consists of four oriented triangles and can be symbolically written as (see Figure 6):  EMBED Equation.DSMT4   MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT ( SEQ MTEqn \c \* Arabic \* MERGEFORMAT 5) The minus signs in this expression indicate orientation reversal: for example, the triangle  EMBED Equation.DSMT4  is on the boundary of the (oriented) tetrahedron  EMBED Equation.DSMT4  and not  EMBED Equation.DSMT4 . Figure 6: Orientation of tetrahedron facets. The orientation of the triangular facets of a tetrahedron according to equation  GOTOBUTTON ZEqnNum186486 \* MERGEFORMAT  REF ZEqnNum186486 \! \* MERGEFORMAT (5) is shown. The order of the vertex IDs in the triplet can be interpreted as the order in which to visit the triangle corners. For example, the red arrow indicates the traversal direction for triangle  EMBED Equation.DSMT4 . Since the starting point of the traversal is arbitrary,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  represent the same oriented triangle (even permutations).  EMBED Equation.DSMT4 , on the other hand, represents the same triangle with the opposite orientation.  EMBED Visio.Drawing.11  According to equation  GOTOBUTTON ZEqnNum186486 \* MERGEFORMAT  REF ZEqnNum186486 \! \* MERGEFORMAT (5), the triangles on the boundary of an oriented tetrahedron inherit their orientation from the tetrahedron. To correctly orient our surface triangles, we must go back to the adjacent tetrahedra and check their orientation. Getting back to the tetrahedra from the set of normalized, un-oriented triplets is a little less straightforward than we would like: no matter which representation of the tetrahedra we use, TetrahedronVertices or TetQuadRep, we cannot link them up directly to sorted triplets. At this point, SQLs UNPIVOT operator comes in handy to (DB-) normalize our triangle set: SELECT TriID, VertexID INTO #triangles FROM ( SELECT ROW_NUMBER() OVER (ORDER BY [0], [1], [2]) AS TriID, [0], [1], [2] FROM ( SELECT A.VertexID AS [0], B.VertexID AS [1], C.VertexID AS [2] FROM TetrahedronVertices AS A JOIN TetrahedronVertices AS B ON A.ElemID = B.ElemID JOIN TetrahedronVertices AS C ON B.ElemID = C.ElemID WHERE A.VertexID < B.VertexID AND B.VertexID < C.VertexID GROUP BY A.VertexID, B.VertexID, C.VertexID HAVING COUNT(A.ElemID) = 1) AS D ) AS E UNPIVOT (VertexID FOR Rank IN ([0], [1], [2])) AS unpvt CREATE INDEX IDX_VertexID ON #triangles(VertexID) All the UNPIVOT operator does is to split each triplet into three rows, one for each vertex. We also used the ROW_NUMBER ranking function to create a temporary triangle identifier. Since we will reuse this set a few times, it is worthwhile creating a temporary table and creating an index on the VertexID column (see below). Joining the #triangles table with the TetrahedronVertices relation gets the orientations right; however, there is one final twist. Different visualization or FEA packages often have different conventions of how to number or rank entities. A tetrahedron has on its boundary four corners, six edges, and four triangles: within each category, which is first, second etc.? This is clearly arbitrary but necessary to make any practical implementation work. We follow the convention used in P. Wawrzyneks FemLib [ REF _Ref127950350 \h 18] library. Let  EMBED Equation.DSMT4  be a tetrahedron. It is FemLibs convention that the four oriented triangles on its boundary are given (in order) by:  EMBED Equation.DSMT4 . For example, the third facet of tetrahedron (12, 4711, 841, 3) would be (841, 3, 12). Since both tetrahedra and triangles are stored in (DB-)normal form, we also need a mapping between tetrahedron local vertex IDs (which range from 0 to 3) and triangle local vertex IDs (which range from 0 to 2). We store the FemLib convention and this mapping in a small lookup table as follows: CREATE TABLE FemLibTetFaces ( TetFaceRank tinyint NOT NULL CHECK(TetFaceRank BETWEEN 0 AND 3), TetVertexRank tinyint NOT NULL CHECK(TetVertexRank BETWEEN 0 AND 3), TriVertexRank tinyint NOT NULL CHECK(TriVertexRank BETWEEN 0 AND 2), CONSTRAINT PK_FemLibTetFaces PRIMARY KEY (TetFaceRank, TetVertexRank) ) The table has only twelve rows (three entries for each of the four faces): TetFaceRankTetVertexRankTriVertexRank000011022110131122220231202300331312 The final query looks as follows: SELECT C.ElemID, TriVertexRank AS Rank, VertexID FROM ( SELECT ElemID, 0 AS TetFaceRank FROM #triangles AS A JOIN TetrahedronVertices AS B ON A.VertexID = B.VertexID WHERE Rank != 3 GROUP BY ElemID, TriID HAVING COUNT(A.VertexID) = 3 UNION ALL SELECT ElemID, 1 AS TetFaceRank FROM #triangles AS A JOIN TetrahedronVertices AS B ON A.VertexID = B.VertexID WHERE Rank != 0 GROUP BY ElemID, TriID HAVING COUNT(A.VertexID) = 3 UNION ALL SELECT ElemID, 2 AS TetFaceRank FROM #triangles AS A JOIN TetrahedronVertices AS B ON A.VertexID = B.VertexID WHERE Rank != 1 GROUP BY ElemID, TriID HAVING COUNT(A.VertexID) = 3 UNION ALL SELECT ElemID, 3 AS TetFaceRank FROM #triangles AS A JOIN TetrahedronVertices AS B ON A.VertexID = B.VertexID WHERE Rank != 2 GROUP BY ElemID, TriID HAVING COUNT(A.VertexID) = 3 ) AS C JOIN FemLibTetFaces AS D ON C.TetFaceRank = D.TetFaceRank JOIN TetrahedronVertices AS E ON C.ElemID = E.ElemID AND D.TetVertexRank = E.Rank On our test machine, for the given mesh, the query runs in about 18 seconds, which brings the total time for the (oriented) surface recovery to around 310 seconds or a little over 5 minutes. Clearly, this solution can be easily generalized for meshes with mixed element types (tets, bricks, wedges, and pyramids) and triangles and quadrilaterals in the surface mesh. For a modest amount of SQL, we get a fast and highly parallel solution!  In this report, we ignore the enormous body of data and knowledge produced by experiments, captured by sensors, and published in the literature. Despite access limitations to parts of this information due to commercial and/or national defense value, modeling and charting just the publicly accessible materials universe remains a formidable task.  FEA experts will have heard by now about mesh-free FE methods. Despite their limited success in solving non-academic problems, the underlying computational geometry problems turn out to be almost identical to the ones encountered in traditional FEA. Problem 3 (interpolation from a cloud of points) described below is a standard problem in the mesh-free arena.  See Appendix B for a non-geometrical sample problem.  An alternative way to establish the inclusion is to look at the signs of the (four) inner products of the inward pointing normals of a tetrahedrons triangular faces with the vectors from the face centroids to  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  is inside, iff all of them are non-negative.  The ray might intersect an edge or a vertex in which case there are multiple choices.  For simplicity, cavities, holes and the like are ignored here. The algorithm can be easily modified to handle those cases [ REF _Ref124912323 \h 7].  The PostgreSQL implementation described in [ REF _Ref124912323 \h 7] does not store the Hilbert codes: the B-tree is built on Hilbert order and an appropriate comparison function is used in the B-tree traversal. SQL Server does not give us this level of control and we have to mimic similar behavior through a clustered index.  We also have not discussed SQL Servers ability to host XML Web service endpoints without IIS.  We ask the mathematically inclined reader to bear with us: the suggested correlation between orientability and two-sidedness of a closed surface is an artifact of the three-dimensional Euclidean space. Orientability is an inner topological property, single- or two-sidedness is not. All of the following do exist: orientable two-sided faces, non-orientable single-sided faces, orientable single-sided faces, and non-orientable two-sided faces. [ REF _Ref128281950 \h 22, REF _Ref128282354 \h 23]  We do not mean normalized in the database sense of normal forms, but unique.  The swap of two successive entries in an n-tuple is called a transposition. For example, (1,3,2,4) is a transposition of (1,2,3,4). It can be shown that an arbitrary permutation of the entries can be represented as a sequence of transpositions. [ REF _Ref128280434 \h 21] A permutation is called even, iff it can be represented as an even number of transpositions. Otherwise it is called odd. For example, (2,3,1) is an even permutation of (1,2,3), whereas (1,3,2) is an odd permutation of (1,2,3).  T-SQL in SQL Server 2005 supports so-called Common Table Expressions (CTE). Though tempting, using a CTE instead of the temporary table is not a good idea (in this example): the index on the vertex ID column of the table is crucial for performance.      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