基于局部摸索的差分进化算法及其在曲面重建中的应用
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摘要
当前差分进化算法及相关的启发性算法被广泛用于求解复杂的非线性优化问题。然而经典的差分进化算法有很多不足之处,如局部搜索能力不够或者加入局部搜索后群体的多样性得不到保证等。针对这些不足之处,本论文对如何设计高效的进化算法来求解复杂的优化问题进行了研究,并将这些算法应用于工程领域两个重要的问题曲面重建与曲面延伸中。本论文的主要工作如下：
在本论文的第二章,设计了两个有效的进化算法分别求解连续函数优化问题和离散函数优化问题。一方面,将经典的梯度法嵌入到差分进化算法中,并设计了一个新的变异算子来保持群体的多样性。用18个典型的连续测试问题,将本文的算法与其它文献中的5个算法进行比较,结果表明本文的算法在寻优能力、平均运行时间和多样性保持等方面都具有明显的优势。另一方面,本文将随机个体之间的差异信息引入差分进化算法中,设计了一个新的二进制进化算法来求解离散优化问题,并证明了该算法的依概率收敛性。数值结果也证明了新算法的有效性。
在本论文的第三章,用非均匀有理B样条函数结合优化算法来重建三维数据的曲面。对一般的不规则点阵(散乱数据),设计了两种点阵初始参数化方法,一种是基于点阵在平面上的投影,另一种采用“分割—参数化”的形式。对五个典型规则点阵的测试问题,精度能达到10-5—10-9；对两组复杂几何曲面上的散乱点阵进行测试,精度能达到10-4；对实测的马山数据精度能达到10--4。同时也考查了噪声对算法的影响,噪声强度分别为1%,5%10%和20%的数值实验结果表明我们设计的算法是鲁棒的。
本文的第四章结合差分进化算法提出了一个一般化的曲面延伸方法,然后将该方法具体化为两个算法分别应用于两个实际问题—孔洞填充与曲面外插。在一个合适选取的截平面上,该方法通过预测点之间的平均距离与转角来获取所要延伸区域的点,针对两个不同的问题,该方法通过约束三角化或重新重建点阵将预测点阵加入原始模型,并且采用光滑化或网格优化操作优化得到局部模型。对球模型和象模型等孔洞填充问题,与最近文献的两个方法在运行时间和预测误差两个方面进行比较,结果表明该算法在精度方面具有优势；对于曲面外插,本文利用抽水蓄能电站—广蓄、马山和白莲河的实测数据来预测水轮水泵机小开度和大开度处的转速、流量和力矩等全特性数据,精度能达到10-2-10-3。
Differential evolution (DE) and the related heuristic algorithms have been widely used to solve complex nonlinear optimization problems for recent years. However, the classical DE exists some shortcomings, such as the limited local search ability or the diversity of the population can not be maintained at an appropriate level. Considering these shortcomings in the DE, this dissertation studied how to design effective evolu-tionary algorithms to solve complex optimization problems, and how to apply these algorithms to solve two important problems in engineering field, that is, surface recon-struction and surface extending. The main works of this dissertation are as follows:
In the second chapter of this dissertation, two effective evolutionary algorithms are proposed to solve continuous and discrete optimization problems, respectively. On one aspect, the classical gradient operator is integrated into the searching strategies of the DE, and a new mutation operator is designed to maintain the diversity of the popu-lation in the evolution procedure. Based on18benchmark continuous test problems, numerical comparisons of the proposed algorithm and five related algorithms in other references show that the proposed algorithm achieves significantly better performance on the searching ability, the average runtime and the population diversity. To solve discrete optimization problems, on the other aspect, the difference information between stochastic individuals are embedded into the DE algorithm and a new binary evolution-ary algorithm is obtained. Furthermore, the convergence of this algorithm in probability is proved and numerical results show the effectiveness of the new algorithm.
In the third chapter of this dissertation, the non-uniform rational B-spline function integrated with optimization algorithms is employed to reconstruct three dimension-al surface. With respect to the general unorganized points (disordered points), two methods for obtaining the initial parameterization are proposed, one is based on the projection of the given points on a plane, the other is to employ the form of "segment-parameterization'". The accuracy on five sets of classical organized points can reach10-5-10-9; for two sets of unorganized points from complex surfaces, the accuracy reaches an order of10-4. Moreover, the accuracy on real-measured data from Mashan pump station can reach10-4. Furthermore, the effect of noise on the algorithm is s- tudied, numerical results on perturbed points with noise intensities of1%.5%10%and20%demonstrate the robustness of the proposed algorithm.
In the fourth chapter of this work, a general method of surface extending by uti-lizing DE algorithm is proposed, which is specialized into two algorithms for two real applications, one is for hole filling, and the other is for surface extrapolation. On a care-fully chosen section plane, this method predicts the points in the considered region by predicting the average distances and intersection angles between the prediction points. With respect, to these two problems, the predicted points are further added into the provided model by constrained triangulation or surface reconstruction, moreover, the obtained local models are improved by smoothing operation or the mesh optimization. On the aspect of hole filling, numerical comparisons of the proposed algorithm and two recently introduced algorithms on the runtime and the prediction accuracy w.r.t. the sphere and elephant models are conducted, and these results demonstrate that our algo-rithm is competitive on the prediction accuracy. On the aspect of surface extrapolation, the proposed algorithm is employed to obtain the complete characteristic data of flow rate, torque and rotational speed at small and large guide vane openings for the pump stations of Guangxu. Mashan and Bailianhe, the prediction accuracies reach10-2-10-3on average.
在本论文的第二章,设计了两个有效的进化算法分别求解连续函数优化问题和离散函数优化问题。一方面,将经典的梯度法嵌入到差分进化算法中,并设计了一个新的变异算子来保持群体的多样性。用18个典型的连续测试问题,将本文的算法与其它文献中的5个算法进行比较,结果表明本文的算法在寻优能力、平均运行时间和多样性保持等方面都具有明显的优势。另一方面,本文将随机个体之间的差异信息引入差分进化算法中,设计了一个新的二进制进化算法来求解离散优化问题,并证明了该算法的依概率收敛性。数值结果也证明了新算法的有效性。
在本论文的第三章,用非均匀有理B样条函数结合优化算法来重建三维数据的曲面。对一般的不规则点阵(散乱数据),设计了两种点阵初始参数化方法,一种是基于点阵在平面上的投影,另一种采用“分割—参数化”的形式。对五个典型规则点阵的测试问题,精度能达到10-5—10-9；对两组复杂几何曲面上的散乱点阵进行测试,精度能达到10-4；对实测的马山数据精度能达到10--4。同时也考查了噪声对算法的影响,噪声强度分别为1%,5%10%和20%的数值实验结果表明我们设计的算法是鲁棒的。
本文的第四章结合差分进化算法提出了一个一般化的曲面延伸方法,然后将该方法具体化为两个算法分别应用于两个实际问题—孔洞填充与曲面外插。在一个合适选取的截平面上,该方法通过预测点之间的平均距离与转角来获取所要延伸区域的点,针对两个不同的问题,该方法通过约束三角化或重新重建点阵将预测点阵加入原始模型,并且采用光滑化或网格优化操作优化得到局部模型。对球模型和象模型等孔洞填充问题,与最近文献的两个方法在运行时间和预测误差两个方面进行比较,结果表明该算法在精度方面具有优势；对于曲面外插,本文利用抽水蓄能电站—广蓄、马山和白莲河的实测数据来预测水轮水泵机小开度和大开度处的转速、流量和力矩等全特性数据,精度能达到10-2-10-3。
Differential evolution (DE) and the related heuristic algorithms have been widely used to solve complex nonlinear optimization problems for recent years. However, the classical DE exists some shortcomings, such as the limited local search ability or the diversity of the population can not be maintained at an appropriate level. Considering these shortcomings in the DE, this dissertation studied how to design effective evolu-tionary algorithms to solve complex optimization problems, and how to apply these algorithms to solve two important problems in engineering field, that is, surface recon-struction and surface extending. The main works of this dissertation are as follows:
In the second chapter of this dissertation, two effective evolutionary algorithms are proposed to solve continuous and discrete optimization problems, respectively. On one aspect, the classical gradient operator is integrated into the searching strategies of the DE, and a new mutation operator is designed to maintain the diversity of the popu-lation in the evolution procedure. Based on18benchmark continuous test problems, numerical comparisons of the proposed algorithm and five related algorithms in other references show that the proposed algorithm achieves significantly better performance on the searching ability, the average runtime and the population diversity. To solve discrete optimization problems, on the other aspect, the difference information between stochastic individuals are embedded into the DE algorithm and a new binary evolution-ary algorithm is obtained. Furthermore, the convergence of this algorithm in probability is proved and numerical results show the effectiveness of the new algorithm.
In the third chapter of this dissertation, the non-uniform rational B-spline function integrated with optimization algorithms is employed to reconstruct three dimension-al surface. With respect to the general unorganized points (disordered points), two methods for obtaining the initial parameterization are proposed, one is based on the projection of the given points on a plane, the other is to employ the form of "segment-parameterization'". The accuracy on five sets of classical organized points can reach10-5-10-9; for two sets of unorganized points from complex surfaces, the accuracy reaches an order of10-4. Moreover, the accuracy on real-measured data from Mashan pump station can reach10-4. Furthermore, the effect of noise on the algorithm is s- tudied, numerical results on perturbed points with noise intensities of1%.5%10%and20%demonstrate the robustness of the proposed algorithm.
In the fourth chapter of this work, a general method of surface extending by uti-lizing DE algorithm is proposed, which is specialized into two algorithms for two real applications, one is for hole filling, and the other is for surface extrapolation. On a care-fully chosen section plane, this method predicts the points in the considered region by predicting the average distances and intersection angles between the prediction points. With respect, to these two problems, the predicted points are further added into the provided model by constrained triangulation or surface reconstruction, moreover, the obtained local models are improved by smoothing operation or the mesh optimization. On the aspect of hole filling, numerical comparisons of the proposed algorithm and two recently introduced algorithms on the runtime and the prediction accuracy w.r.t. the sphere and elephant models are conducted, and these results demonstrate that our algo-rithm is competitive on the prediction accuracy. On the aspect of surface extrapolation, the proposed algorithm is employed to obtain the complete characteristic data of flow rate, torque and rotational speed at small and large guide vane openings for the pump stations of Guangxu. Mashan and Bailianhe, the prediction accuracies reach10-2-10-3on average.
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