论文标题
快速计算线性振动系统的最佳阻尼参数
Fast computation of optimal damping parameters for linear vibrational systems
论文作者
论文摘要
我们制定了线性振动系统数学模型的基础二次特征问题，作为对角线 - 加上级矩阵$ a $的特征值问题。 $ a $的特征向量矩阵具有类似cauchy的结构。最佳粘度是$ trace（x）$的最小值，其中$ x $是lyapunov方程的解决方案$ ax+ax+xa^{*} = gg^{*} $。在这里，$ g $是一个低级别矩阵，取决于需要抑制的特征频率。在需要$ o（n^3）$操作的线性化问题的初始特征值分解之后，我们的算法为每种选择外部阻尼器在$ O（n^2）$操作中计算最佳粘度，但前提是dampers的数量很小。因此，随后的优化比在每个步骤中求解Lyapunov方程的标准方法中的数量级快，因此需要$ O（n^3）$操作。我们的算法基于$ O（n^2）$ eigensolver用于复杂的对称对角线级矩阵和快速$ o（n^2）$ linked cauchy样矩阵的乘法。
We formulate the quadratic eigenvalue problem underlying the mathematical model of a linear vibrational system as an eigenvalue problem of a diagonal-plus-low-rank matrix $A$. The eigenvector matrix of $A$ has a Cauchy-like structure. Optimal viscosities are those for which $trace(X)$ is minimal, where $X$ is the solution of the Lyapunov equation $AX+XA^{*}=GG^{*}$. Here $G$ is a low-rank matrix which depends on the eigenfrequencies that need to be damped. After initial eigenvalue decomposition of linearized problem which requires $O(n^3)$ operations, our algorithm computes optimal viscosities for each choice of external dampers in $O(n^2)$ operations, provided that the number of dampers is small. Hence, the subsequent optimization is order of magnitude faster than in the standard approach which solves Lyapunov equation in each step, thus requiring $O(n^3)$ operations. Our algorithm is based on $O(n^2)$ eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast $O(n^2)$ multiplication of linked Cauchy-like matrices.
