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    • 2. 发明申请
    • DETERMINING FIELD-DEPENDENT CHARACTERISTICS BY EMPLOYING HIGH-ORDER QUADRATURES IN THE PRESENCE OF GEOMETRIC SINGULARITIES
    • 在存在几何奇异性的情况下通过使用高阶求和确定场相关特性
    • WO2004023352A3
    • 2004-08-05
    • PCT/US0328229
    • 2003-09-08
    • MAD MAX OPTICS INCCRUTCHFIELD WILLIAM YETHRIDGE JOSEPH FRANKLINGIMBUTAS ZYDRUNASGREENGARD LESLIE FROKHLIN VLADIMIR
    • CRUTCHFIELD WILLIAM YETHRIDGE JOSEPH FRANKLINGIMBUTAS ZYDRUNASGREENGARD LESLIE FROKHLIN VLADIMIR
    • G06F17/13G06F17/50
    • G06F17/13G06F17/5018
    • A machine for determining field-dependent physical characteristics contains tables of precomputed quadratures and employs them to integrate numerically over a problem boundary. The quadratures are based on products of a kernel function and a basis s that spans a wide range of density functions. The kernel function is dependent on a target node's position, and different quadratures are precomputed for different target-node positions or ranges thereof. In the case of at least some of the quadratures, some the basis functions include integrable singularities. The solver divides the problem boundary into a plurality of problem intervals, to which it maps the canonical interval. To integrate a problem interval for a target point, the solver employs a precomputed quadrature that is associated with the target point's relative position and that was generated by using a basis in which a singularity occurs at each canonical-­interval location that was mapped to a geometrical singularity on the problem interval. The quadrature results in high-­order accuracy even if no individual basis function includes a singularity whose shape is the same as one induced by the geometric singularity. These quadratures can be coupled with a Fast Multipole Method ("FMM") to evaluate layer potentials rapidly and with high accuracy.
    • 用于确定依赖于场的物理特性的机器包含预先计算的正方形表格,并使用它们在问题边界上进行数值积分。 这些quadratures基于核函数的产品和跨越各种密度函数的基础。 核函数依赖于目标节点的位置,并且针对不同的目标节点位置或其范围预先计算不同的正交矩阵。 在至少一些quadratures的情况下,一些基函数包括可积奇点。 求解器将问题边界划分为多个问题区间,并将其映射到规范区间。 为了整合目标点的问题区间,求解器使用与目标点的相对位置相关的预先计算的正交并且是通过使用在每个正则间隔位置出现奇点的基础被映射到几何 问题区间的奇点。 即使没有单独的基函数包含奇点,其形状与由几何奇点引起的奇点相同,正交仍然导致高精度。 这些quadratures可以结合快速多极方法(“FMM”)来快速和高精度地评估层电位。
    • 6. 发明申请
    • DETERMINING FIELD-DEPENDENT CHARACTERISTICS BY EMPLOYING HIGH-ORDER QUADRATURES IN THE PRESENCE OF GEOMETRIC SINGULARITIES
    • 通过在几何形状存在下采用高阶准则来确定场依赖特性
    • WO2004023352A2
    • 2004-03-18
    • PCT/US2003/028229
    • 2003-09-08
    • MAD MAX OPTICS, INC.CRUTCHFIELD, William, Y.ETHRIDGE, Joseph, FranklinGIMBUTAS, ZydrunasGREENGARD, Leslie, F.ROKHLIN, Vladimir
    • CRUTCHFIELD, William, Y.ETHRIDGE, Joseph, FranklinGIMBUTAS, ZydrunasGREENGARD, Leslie, F.ROKHLIN, Vladimir
    • G06F17/50
    • G06F17/13G06F17/5018
    • A machine for determining field-dependent physical characteristics contains tables of precomputed quadratures and employs them to integrate numerically over a problem boundary. The quadratures are based on products of a kernel function and a basis s that spans a wide range of density functions. The kernel function is dependent on a target node's position, and different quadratures are precomputed for different target-node positions or ranges thereof. In the case of at least some of the quadratures, some the basis functions include integrable singularities. The solver divides the problem boundary into a plurality of problem intervals, to which it maps the canonical interval. To integrate a problem interval for a target point, the solver employs a precomputed quadrature that is associated with the target point's relative position and that was generated by using a basis in which a singularity occurs at each canonical-­interval location that was mapped to a geometrical singularity on the problem interval. The quadrature results in high-­order accuracy even if no individual basis function includes a singularity whose shape is the same as one induced by the geometric singularity. These quadratures can be coupled with a Fast Multipole Method ("FMM") to evaluate layer potentials rapidly and with high accuracy.
    • 用于确定场相关物理特征的机器包含预计算正交表的表,并使用它们在问题边界上数字积分。 正交基于核函数的产物和跨越各种密度函数的基础。 内核函数取决于目标节点的位置,并且针对不同的目标节点位置或其范围预先计算不同的正交。 在至少一些正交的情况下,一些基函数包括可积分奇点。 求解器将问题边界划分成多个问题间隔,它映射规范间隔。 为了整合目标点的问题间隔,求解器使用与目标点的相对位置相关联的预计算正交,并且通过使用在映射到几何的每个规范间隔位置处发生奇点的基础来生成 关于问题间隔的奇点。 即使没有单独的基函数包括其形状与由几何奇点引起的形状相同的奇异点,正交也导致高阶精度。 这些正交可以与快速多极方法(“FMM”)耦合,以高精度地快速评估层电位。