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Design of C++ Matrix Library for FEM Computing Base on Generic Programming

Design of C++ Matrix Library for FEM Computing Base on Generic Programming. Zhou Jianhui. Existing linear algebra library(1). Existing linear algebra library(2). Summary:

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Design of C++ Matrix Library for FEM Computing Base on Generic Programming

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  1. Design of C++ Matrix Library for FEM Computing Base on Generic Programming Zhou Jianhui

  2. Existing linear algebra library(1)

  3. Existing linear algebra library(2) Summary: 1. C++ code is by no means inherently slower than Fortran code. After being optimized, C++ matrix library perform equally or even better than that Fortran in term of corresponding codes. 2. Almost all high-performance C++ matrix library use generic programming technique to optimize performance. Generic programming is proved to be of key importance to high-performance numerical computation. 3. C++ matrix libraries have more convenient interface. The operator overloading of C++ makes it possible to write program as easy as writing algebra expressions. For example, if you do matrix addition operation, you can write: A = B + C, rather than: call addition (A,B,C).

  4. Overview As the new C++ matrix library hereby is oriented towards FEM computing, it only provides operations of dense, symmetric and banded matrix, vector, sub-matrix, etc, rather than that of all algebra operations. But it does provide optimization specifically for FEM computing optimizing. The library was developed with references to those existing matrix libraries mentioned above, which brings some similarities between them resulting from the same optimizing technique based on generic programming. Notably, the library has made some significant improvements, including a novel architecture, an upgraded optimizing technique and a efficient memory management policy. The library achieves better performance, user interface and capability for further expansion.

  5. Architecture

  6. User Interface(1) Some of the user interface

  7. User Interface(2) Some of the user interface

  8. User Interface(3) Some of the user interface

  9. The realization of optimizing by C++ generic programming. Two of the most widely used optimizing technique are: • Expression templates. • Template metaprograms.

  10. Expression templates: eliminate temporary matrix Matrix addition: A = B + C Fortran code will act as follows: while C++ codes usually generate temporary variables implicitly and actually act as follows: Such temporary matrix occupies too much spaces and is time consuming. Such drawbacks can be eliminated by expression templates, making C++ code as efficient as Fortran Code.

  11. Template metaprograms: minimize cost of loop Inner production of vectors : dot = V1 * V2, suppose size of V1,V2 are 3 General codes can be depicted as: Float dot = 0; DOT = 0 For(int i = 0; i < 3; ++i) DO 100 I = 1, 3 dot += v1[i] * v2[i]; 100 DOT = DOT + V1(I) * V2(I) But with the size of two vectors given in advance, the code can be more efficient as follows: dot = v1[0] * v2[0] + v1[1] * v2[1] + v1[2] * v2[2]; or: DOT = V1(1) * V2(1) + V1(2) * V2(2) + V1(3) * V2(3) Essential difference between them is that the latter codes saves the runtime of loop,such as increment of looping variant. With template metaprograms, C++ can generate codes like the latter form automatically during compiling process on condition that size of matrix or vector is known beforehand.

  12. The highlight of the new matrix library 1. Improved implementation of expression templates, which further simplifies the user interface. 2. Efficient memory management. The library does memory management itself instead of the operating system. 3. New library architecture, as mentioned before. 4. Some specific optimization for FEM computing. (1) Special storage strategy and the corresponding algorithms for special matrix, such as stiffness matrix. (2) Use template metaprograms as frequently as possible.

  13. Specific optimizing for FEM computing (1) Specific storage strategy and the corresponding algorithms for specific matrix Example: In FEM computing, as K is a symmetric matrix, thus the multiplication algorithm can be optimized so that its calculation can be reduced by half

  14. FEM computing-specific optimizing (2) Use template metaprograms as widely as possible. Example: For an eight-node isoparametric element The size of all the above matrixes are fixed, this is exactly where template metaprograms can be used to optimize performance. Actually, there are so many matrix operations where matrix and vector are size-fixed, that template metaprograms can be frequently used to achieve high performance.

  15. Performance Comparison The program is compiled under Visual C++7.1, and run on Intel PⅢ550E machine for 500000 times. For instance, the matrix of the isoparametric element operations in the program can be depicted as follows:

  16. Thanks for listening！

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