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C++实现有限元计算 矩阵装配Assembly类

article 2025/1/27 12:38:39

本系列文章致力于实现“手搓有限元，干翻Ansys的目标”，基本框架为前端显示使用QT实现交互，后端计算采用Visual Studio C++。

1、Assembly

Assembly类用来实现矩阵的装配，将单元的全局刚度矩阵、位移列矩阵、外载列矩阵装配组装为整体全局刚度矩阵、位移列矩阵和外载列矩阵，作为后续求解位移列阵的基础。

Assembly类组成架构

1.2、public function

1.2.1、构造函数与析构函数

矩阵装配过程中不需要进行初始化，所以构造函数与析构函数只做定义，不做实现（空实现）。

Assembly.h函数声明：

//***********************构造函数析构函数***********************//
/*
函数名称:		无参构造函数
*/
Assembly();

/*
函数名称:		析构函数
*/
~Assembly();

Assembly.cpp函数实现：（空实现）

1.2.2、单元全局刚度矩阵组装

将单元的全局刚度矩阵装配为整体全局刚度矩阵，单元包括上一期的二维杆单元Bar2D2Node（点击此处跳转Bar2D2Node介绍）、以及将来会介绍的三维杆单元Bar3D2Node、二维截面为矩形的梁单元Rect_Beam2D2Node等等。

Assembly.h函数声明：

//***************************组装矩阵***************************//
/*
刚度矩阵的组装Bar2D2Node
bar:				二维杆单元数组
numPoint:			节点数量
numElement:			单元数量
*/
Matrix AssemblyMatK(std::vector<Bar2D2Node>bar, int numPoint, int numElement);

/*
刚度矩阵的组装Bar3D2Node
bar:				三维杆单元数组
numPoint:			节点数量
numElement:			单元数量
*/
Matrix AssemblyMatK(std::vector<Bar3D2Node>bar, int numPoint, int numElement);

/*
刚度矩阵的组装Rect_Beam2D2Node
beam:				二维矩形截面梁单元数组
numPoint:			节点数量
numElement:			单元数量
*/
Matrix AssemblyMatK(std::vector<Rect_Beam2D2Node>beam, int numPoint, int numElement);

Assembly.cpp函数实现：

//刚度矩阵的组装Bar2D2Node
Matrix Assembly::AssemblyMatK(std::vector<Bar2D2Node>bar, int numPoint, int numElement)
{
	//建立组装刚度矩阵 由于是2D问题 矩阵维度为2*numPoint
	this->m_SumK.CopyMat(Matrix(2 * numPoint, 2 * numPoint));

	for (int iEle = 0; iEle < numElement; iEle++)
	{
		//获取点ID
		int pointID0 = bar[iEle].GetPoint0()->GetID();
		int pointID1 = bar[iEle].GetPoint1()->GetID();

		//ID List [2i, 2i+1, 2j, 2j+1]
		int pointIndex[4] = { 2 * pointID0, 2 * pointID0 + 1, 2* pointID1, 2* pointID1 + 1};

		//2D问题 逐个组装刚度矩阵
		for (int iRow = 0; iRow < 4; iRow++)
		{
			for (int iCol = 0; iCol < 4; iCol++)
			{
				double Kele = bar[iEle].GetK().GetMatrixEle(iRow, iCol) + this->m_SumK.GetMatrixEle(pointIndex[iRow], pointIndex[iCol]);
				this->m_SumK.SetMatrixEle(pointIndex[iRow], pointIndex[iCol], Kele);
			}
		}
	}

	return this->m_SumK;
}

//刚度矩阵的组装Bar3D2Node
Matrix Assembly::AssemblyMatK(std::vector<Bar3D2Node>bar, int numPoint, int numElement)
{
	//建立组装刚度矩阵 由于是3D问题 矩阵维度为3*numPoint
	this->m_SumK.CopyMat(Matrix(3 * numPoint, 3 * numPoint));

	for (int iEle = 0; iEle < numElement; iEle++)
	{
		//获取点ID
		int pointID0 = bar[iEle].GetPoint0()->GetID();
		int pointID1 = bar[iEle].GetPoint1()->GetID();

		//ID List [3i, 3i+1, 3i+2, 3j, 3j+1, 3j+2]
		int pointIndex[6] = { 3 * pointID0 + 0, 3 * pointID0 + 1, 3 * pointID0 + 2,
			3 * pointID1 + 0, 3 * pointID1 + 1, 3 * pointID1 + 2};

		//3D问题 逐个组装刚度矩阵
		for (int iRow = 0; iRow < 6; iRow++)
		{
			for (int iCol = 0; iCol < 6; iCol++)
			{
				//刚度矩阵累加
				double Kele = bar[iEle].GetK().GetMatrixEle(iRow, iCol) + this->m_SumK.GetMatrixEle(pointIndex[iRow], pointIndex[iCol]);

				//矩阵元素装填
				this->m_SumK.SetMatrixEle(pointIndex[iRow], pointIndex[iCol], Kele);
			}
		}
	}

	return this->m_SumK;
}

//刚度矩阵的组装Rect_Beam2D2Node
Matrix Assembly::AssemblyMatK(std::vector<Rect_Beam2D2Node>beam, int numPoint, int numElement)
{
	//建立组装刚度矩阵 由于是2D问题 矩形梁每个节点自由度为3 矩阵维度为3*numPoint
	this->m_SumK.CopyMat(Matrix(3 * numPoint, 3 * numPoint));

	for (int iEle = 0; iEle < numElement; iEle++)
	{
		//获取点ID
		int pointID0 = beam[iEle].GetPoint0()->GetID();
		int pointID1 = beam[iEle].GetPoint1()->GetID();

		//ID List [3i, 3i+1, 3i+2, 3j, 3j+1, 3j+2]
		int pointIndex[6] = { 3 * pointID0 + 0, 3 * pointID0 + 1, 3 * pointID0 + 2,
			3 * pointID1 + 0, 3 * pointID1 + 1, 3 * pointID1 + 2 };

		//3D问题 逐个组装刚度矩阵
		for (int iRow = 0; iRow < 6; iRow++)
		{
			for (int iCol = 0; iCol < 6; iCol++)
			{
				//刚度矩阵累加
				double Kele = beam[iEle].GetK().GetMatrixEle(iRow, iCol) + this->m_SumK.GetMatrixEle(pointIndex[iRow], pointIndex[iCol]);

				//矩阵元素装填
				this->m_SumK.SetMatrixEle(pointIndex[iRow], pointIndex[iCol], Kele);
			}
		}
	}

	return this->m_SumK;
}

1.2.3、位移列矩阵组装

将各个单元的节点位移装配为整体节点位移。注意这里只区分为二维节点Point2D（点击此处可跳转到Point2D的介绍）和三维节点Point3D。因为对于不同单元组成的节点无非就二维节点与三维节点两种情况。

Assembly.h函数声明：

/*
线位移、角位移-矩阵组装Point2D
p:					二维节点数组
numPoint:			节点数量
ElementStyle:		单元类型枚举
*/
Matrix AssemblyMatU(std::vector<Point2D> p, int numPoint, ElementStyle eleStyle);

/*
线位移、角位移-矩阵组装Point3D
p:					三维节点数组
numPoint:			节点数量
ElementStyle:		单元类型枚举
*/
Matrix AssemblyMatU(std::vector<Point3D> p, int numPoint, ElementStyle eleStyle);

Assembly.cpp函数实现：

//线位移、角位移 - 矩阵组装Point2D
Matrix Assembly::AssemblyMatU(std::vector<Point2D> p, int numPoint, ElementStyle eleStyle)
{
	//创建结果矩阵
	Matrix* resMat;

	switch (eleStyle)
	{
	case BAR_2D2NODE:

		//位移整体矩阵 2D 两个自由度(杆单元 存在两个自由度)
		resMat = new Matrix(2 * numPoint, 1);

		//遍历节点
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(2 * i + 0, 0, p[i].GetU());			//U
			resMat->SetMatrixEle(2 * i + 1, 0, p[i].GetV());			//V
		}

		break;

	case RECT_BEAM_2D2NODE:

		//位移整体矩阵 2D 三个自由度(梁单元 存在弯曲变形)
		resMat = new Matrix(3 * numPoint, 1);

		//遍历节点
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(3 * i + 0, 0, p[i].GetU());			//U
			resMat->SetMatrixEle(3 * i + 1, 0, p[i].GetV());			//V
			resMat->SetMatrixEle(3 * i + 2, 0, p[i].GetTheatZ());		//TheatZ
		}

		break;

	default:
		resMat = new Matrix(3 * numPoint, 1);
		PRINTF_LOCAL;
		break;

	}
		

	return *resMat;
}

//线位移、角位移-矩阵组装Point3D
Matrix Assembly::AssemblyMatU(std::vector<Point3D> p, int numPoint, ElementStyle eleStyle)
{
	//创建结果矩阵
	Matrix* resMat;

	switch (eleStyle)
	{
	case BAR_3D2NODE:

		//位移整体矩阵 3D
		resMat = new Matrix(3 * numPoint, 1);

		//遍历节点 逐个填取位移数据
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(3 * i + 0, 0, p[i].GetU());			//U
			resMat->SetMatrixEle(3 * i + 1, 0, p[i].GetV());			//V
			resMat->SetMatrixEle(3 * i + 2, 0, p[i].GetW());			//W
		}
		break;

	default:
		resMat = new Matrix(3 * numPoint, 1);
		PRINTF_LOCAL;
		break;
	}

	return *resMat;
}

1.2.4、外载列矩阵组装

将各个单元的节点外载装配为整体节点外载。

Assembly.h函数声明：

/*
力、力矩-矩阵组装Point2D
p:					二维节点数组
numPoint:			节点数量
ElementStyle:		单元类型枚举
*/
Matrix AssemblyMatF(std::vector<Point2D> p, int numPoint, ElementStyle eleStyle);

/*
力、力矩-矩阵组装Point3D
p:					三维节点数组
numPoint:			节点数量
ElementStyle:		单元类型枚举
*/
Matrix AssemblyMatF(std::vector<Point3D> p, int numPoint, ElementStyle eleStyle);

Assembly.cpp函数实现：

//力、力矩-矩阵组装Point2D
Matrix Assembly::AssemblyMatF(std::vector<Point2D> p, int numPoint, ElementStyle eleStyle)
{
	Matrix* resMat;

	switch (eleStyle)
	{
	case BAR_2D2NODE:

		//结果矩阵 2D 两个自由度
		resMat = new Matrix(2 * numPoint, 1);

		//遍历节点
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(2 * i + 0, 0, p[i].GetFx());			//U
			resMat->SetMatrixEle(2 * i + 1, 0, p[i].GetFy());			//V
		}

		break;

	case RECT_BEAM_2D2NODE:

		//结果矩阵 2D 三个自由度
		resMat = new Matrix(3 * numPoint, 1);

		//遍历节点
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(3 * i + 0, 0, p[i].GetFx());			//U
			resMat->SetMatrixEle(3 * i + 1, 0, p[i].GetFy());			//V
			resMat->SetMatrixEle(3 * i + 2, 0, p[i].GetMz());			//Mz
		}

		break;

	default:
		resMat = new Matrix(3 * numPoint, 1);
		PRINTF_LOCAL;
		break;
	}

	return *resMat;
}

//力、力矩-矩阵组装Point3D
Matrix Assembly::AssemblyMatF(std::vector<Point3D> p, int numPoint, ElementStyle eleStyle)
{
	//创建结果矩阵
	Matrix* resMat;

	switch (eleStyle)
	{
	case BAR_3D2NODE:

		//结果矩阵 3D
		resMat = new Matrix(3 * numPoint, 1);

		//遍历节点
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(3 * i + 0, 0, p[i].GetFx());			//x方向外载
			resMat->SetMatrixEle(3 * i + 1, 0, p[i].GetFy());			//y方向外载
			resMat->SetMatrixEle(3 * i + 2, 0, p[i].GetFz());			//z方向外载
		}
		break;

	default:
		resMat = new Matrix(3 * numPoint, 1);
		break;
	}

	return *resMat;
}

1.3、private variable

私有变量，只能在类内进行调用。

private:
	Matrix m_SumK;			//组装后的刚度矩阵

1.4、全部源码

Assembly.h函数声明：

#ifndef _ASSEMBLY_H_
#define _ASSEMBLY_H_

#include "Bar2D2Node.h"
#include "Bar3D2Node.h"
#include "Beam2D2Node.h"
#include "Rectangle_Beam2D2Node.h"
#include "GlobalVars.h"
#include <vector>

class Assembly
{
public:
	//***********************构造函数析构函数***********************//
	/*
	函数名称:		无参构造函数
	*/
	Assembly();

	/*
	函数名称:		析构函数
	*/
	~Assembly();

	//***************************组装矩阵***************************//
	/*
	刚度矩阵的组装Bar2D2Node
	bar:				二维杆单元数组
	numPoint:			节点数量
	numElement:			单元数量
	*/
	Matrix AssemblyMatK(std::vector<Bar2D2Node>bar, int numPoint, int numElement);

	/*
	刚度矩阵的组装Bar3D2Node
	bar:				三维杆单元数组
	numPoint:			节点数量
	numElement:			单元数量
	*/
	Matrix AssemblyMatK(std::vector<Bar3D2Node>bar, int numPoint, int numElement);

	/*
	刚度矩阵的组装Rect_Beam2D2Node
	beam:				二维矩形截面梁单元数组
	numPoint:			节点数量
	numElement:			单元数量
	*/
	Matrix AssemblyMatK(std::vector<Rect_Beam2D2Node>beam, int numPoint, int numElement);

	/*
	线位移、角位移-矩阵组装Point2D
	p:					二维节点数组
	numPoint:			节点数量
	ElementStyle:		单元类型枚举
	*/
	Matrix AssemblyMatU(std::vector<Point2D> p, int numPoint, ElementStyle eleStyle);

	/*
	线位移、角位移-矩阵组装Point3D
	p:					三维节点数组
	numPoint:			节点数量
	ElementStyle:		单元类型枚举
	*/
	Matrix AssemblyMatU(std::vector<Point3D> p, int numPoint, ElementStyle eleStyle);

	/*
	力、力矩-矩阵组装Point2D
	p:					二维节点数组
	numPoint:			节点数量
	ElementStyle:		单元类型枚举
	*/
	Matrix AssemblyMatF(std::vector<Point2D> p, int numPoint, ElementStyle eleStyle);

	/*
	力、力矩-矩阵组装Point3D
	p:					三维节点数组
	numPoint:			节点数量
	ElementStyle:		单元类型枚举
	*/
	Matrix AssemblyMatF(std::vector<Point3D> p, int numPoint, ElementStyle eleStyle);

private:
	Matrix m_SumK;			//组装后的刚度矩阵
};

#endif

Assembly.cpp函数实现：

#include "Assembly.h"

Assembly::Assembly()
{
}

Assembly::~Assembly()
{
}

//刚度矩阵的组装Bar2D2Node
Matrix Assembly::AssemblyMatK(std::vector<Bar2D2Node>bar, int numPoint, int numElement)
{
	//建立组装刚度矩阵 由于是2D问题 矩阵维度为2*numPoint
	this->m_SumK.CopyMat(Matrix(2 * numPoint, 2 * numPoint));

	for (int iEle = 0; iEle < numElement; iEle++)
	{
		//获取点ID
		int pointID0 = bar[iEle].GetPoint0()->GetID();
		int pointID1 = bar[iEle].GetPoint1()->GetID();

		//ID List [2i, 2i+1, 2j, 2j+1]
		int pointIndex[4] = { 2 * pointID0, 2 * pointID0 + 1, 2* pointID1, 2* pointID1 + 1};

		//2D问题 逐个组装刚度矩阵
		for (int iRow = 0; iRow < 4; iRow++)
		{
			for (int iCol = 0; iCol < 4; iCol++)
			{
				double Kele = bar[iEle].GetK().GetMatrixEle(iRow, iCol) + this->m_SumK.GetMatrixEle(pointIndex[iRow], pointIndex[iCol]);
				this->m_SumK.SetMatrixEle(pointIndex[iRow], pointIndex[iCol], Kele);
			}
		}
	}

	return this->m_SumK;
}

//刚度矩阵的组装Bar3D2Node
Matrix Assembly::AssemblyMatK(std::vector<Bar3D2Node>bar, int numPoint, int numElement)
{
	//建立组装刚度矩阵 由于是3D问题 矩阵维度为3*numPoint
	this->m_SumK.CopyMat(Matrix(3 * numPoint, 3 * numPoint));

	for (int iEle = 0; iEle < numElement; iEle++)
	{
		//获取点ID
		int pointID0 = bar[iEle].GetPoint0()->GetID();
		int pointID1 = bar[iEle].GetPoint1()->GetID();

		//ID List [3i, 3i+1, 3i+2, 3j, 3j+1, 3j+2]
		int pointIndex[6] = { 3 * pointID0 + 0, 3 * pointID0 + 1, 3 * pointID0 + 2,
			3 * pointID1 + 0, 3 * pointID1 + 1, 3 * pointID1 + 2};

		//3D问题 逐个组装刚度矩阵
		for (int iRow = 0; iRow < 6; iRow++)
		{
			for (int iCol = 0; iCol < 6; iCol++)
			{
				//刚度矩阵累加
				double Kele = bar[iEle].GetK().GetMatrixEle(iRow, iCol) + this->m_SumK.GetMatrixEle(pointIndex[iRow], pointIndex[iCol]);

				//矩阵元素装填
				this->m_SumK.SetMatrixEle(pointIndex[iRow], pointIndex[iCol], Kele);
			}
		}
	}

	return this->m_SumK;
}

//刚度矩阵的组装Rect_Beam2D2Node
Matrix Assembly::AssemblyMatK(std::vector<Rect_Beam2D2Node>beam, int numPoint, int numElement)
{
	//建立组装刚度矩阵 由于是2D问题 矩形梁每个节点自由度为3 矩阵维度为3*numPoint
	this->m_SumK.CopyMat(Matrix(3 * numPoint, 3 * numPoint));

	for (int iEle = 0; iEle < numElement; iEle++)
	{
		//获取点ID
		int pointID0 = beam[iEle].GetPoint0()->GetID();
		int pointID1 = beam[iEle].GetPoint1()->GetID();

		//ID List [3i, 3i+1, 3i+2, 3j, 3j+1, 3j+2]
		int pointIndex[6] = { 3 * pointID0 + 0, 3 * pointID0 + 1, 3 * pointID0 + 2,
			3 * pointID1 + 0, 3 * pointID1 + 1, 3 * pointID1 + 2 };

		//3D问题 逐个组装刚度矩阵
		for (int iRow = 0; iRow < 6; iRow++)
		{
			for (int iCol = 0; iCol < 6; iCol++)
			{
				//刚度矩阵累加
				double Kele = beam[iEle].GetK().GetMatrixEle(iRow, iCol) + this->m_SumK.GetMatrixEle(pointIndex[iRow], pointIndex[iCol]);

				//矩阵元素装填
				this->m_SumK.SetMatrixEle(pointIndex[iRow], pointIndex[iCol], Kele);
			}
		}
	}

	return this->m_SumK;
}

//线位移、角位移 - 矩阵组装Point2D
Matrix Assembly::AssemblyMatU(std::vector<Point2D> p, int numPoint, ElementStyle eleStyle)
{
	//创建结果矩阵
	Matrix* resMat;

	switch (eleStyle)
	{
	case BAR_2D2NODE:

		//位移整体矩阵 2D 两个自由度(杆单元 存在两个自由度)
		resMat = new Matrix(2 * numPoint, 1);

		//遍历节点
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(2 * i + 0, 0, p[i].GetU());			//U
			resMat->SetMatrixEle(2 * i + 1, 0, p[i].GetV());			//V
		}

		break;

	case RECT_BEAM_2D2NODE:

		//位移整体矩阵 2D 三个自由度(梁单元 存在弯曲变形)
		resMat = new Matrix(3 * numPoint, 1);

		//遍历节点
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(3 * i + 0, 0, p[i].GetU());			//U
			resMat->SetMatrixEle(3 * i + 1, 0, p[i].GetV());			//V
			resMat->SetMatrixEle(3 * i + 2, 0, p[i].GetTheatZ());		//TheatZ
		}

		break;

	default:
		resMat = new Matrix(3 * numPoint, 1);
		PRINTF_LOCAL;
		break;

	}
		

	return *resMat;
}

//线位移、角位移-矩阵组装Point3D
Matrix Assembly::AssemblyMatU(std::vector<Point3D> p, int numPoint, ElementStyle eleStyle)
{
	//创建结果矩阵
	Matrix* resMat;

	switch (eleStyle)
	{
	case BAR_3D2NODE:

		//位移整体矩阵 3D
		resMat = new Matrix(3 * numPoint, 1);

		//遍历节点 逐个填取位移数据
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(3 * i + 0, 0, p[i].GetU());			//U
			resMat->SetMatrixEle(3 * i + 1, 0, p[i].GetV());			//V
			resMat->SetMatrixEle(3 * i + 2, 0, p[i].GetW());			//W
		}
		break;

	default:
		resMat = new Matrix(3 * numPoint, 1);
		PRINTF_LOCAL;
		break;
	}

	return *resMat;
}

//力、力矩-矩阵组装Point2D
Matrix Assembly::AssemblyMatF(std::vector<Point2D> p, int numPoint, ElementStyle eleStyle)
{
	Matrix* resMat;

	switch (eleStyle)
	{
	case BAR_2D2NODE:

		//结果矩阵 2D 两个自由度
		resMat = new Matrix(2 * numPoint, 1);

		//遍历节点
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(2 * i + 0, 0, p[i].GetFx());			//U
			resMat->SetMatrixEle(2 * i + 1, 0, p[i].GetFy());			//V
		}

		break;

	case RECT_BEAM_2D2NODE:

		//结果矩阵 2D 三个自由度
		resMat = new Matrix(3 * numPoint, 1);

		//遍历节点
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(3 * i + 0, 0, p[i].GetFx());			//U
			resMat->SetMatrixEle(3 * i + 1, 0, p[i].GetFy());			//V
			resMat->SetMatrixEle(3 * i + 2, 0, p[i].GetMz());			//Mz
		}

		break;

	default:
		resMat = new Matrix(3 * numPoint, 1);
		PRINTF_LOCAL;
		break;
	}

	return *resMat;
}

//力、力矩-矩阵组装Point3D
Matrix Assembly::AssemblyMatF(std::vector<Point3D> p, int numPoint, ElementStyle eleStyle)
{
	//创建结果矩阵
	Matrix* resMat;

	switch (eleStyle)
	{
	case BAR_3D2NODE:

		//结果矩阵 3D
		resMat = new Matrix(3 * numPoint, 1);

		//遍历节点
		for (int i = 0; i < numPoint; i++)
		{
			resMat->SetMatrixEle(3 * i + 0, 0, p[i].GetFx());			//x方向外载
			resMat->SetMatrixEle(3 * i + 1, 0, p[i].GetFy());			//y方向外载
			resMat->SetMatrixEle(3 * i + 2, 0, p[i].GetFz());			//z方向外载
		}
		break;

	default:
		resMat = new Matrix(3 * numPoint, 1);
		break;
	}

	return *resMat;
}


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