用C++写一段计算偏流的代码

偏流通常是计算机图形学中的术语，在计算流体动力学(CFD)中用于描述流体通过物体表面的速度分量。在C++编写这样的计算通常涉及到向量运算和数值方法。下面是一个简单的例子，假设我们有一个二维网格系统，每个网格点有速度向量v，并需要计算它沿着某个边界方向的偏流。这里用到了基本的向量化操作：

#include <iostream>
#include <vector>

struct Vector2D {
    double x, y;
    Vector2D(double x = 0, double y = 0) : x(x), y(y) {}
};

double dotProduct(Vector2D a, Vector2D b) {
    return a.x * b.x + a.y * b.y;
}

Vector2D normalAtBoundary(const std::vector<Vector2D>& boundaryPoints, int pointIndex) {
    Vector2D n = (boundaryPoints[pointIndex+1] - boundaryPoints[pointIndex]) / norm(boundaryPoints[pointIndex+1] - boundaryPoints[pointIndex]);
    // normalize the vector if needed
    return n;
}

double fluxThroughBoundary(const Vector2D& v, const Vector2D& n, double width) {
    // width is the distance between the two points defining the boundary segment
    return v.x * n.y - v.y * n.x; // cross product as signed area of parallelogram
}

int main() {
    std::vector<Vector2D> grid(10, Vector2D()); // sample 2D grid with some speed vectors v[i]
    std::vector<Vector2D> boundaryPoints(4); // assuming a rectangle, fill with endpoints

    for (int i = 0; i < grid.size(); ++i) {
        Vector2D n = normalAtBoundary(boundaryPoints, (i % boundaryPoints.size())); // get normal at each grid point's boundary
        double flux = fluxThroughBoundary(grid[i], n, 1); // assume uniform grid spacing
        std::cout << "Grid point " << i << " has a flow flux of: " << flux << std::endl;
    }

    return 0;
}