frictional model and umat

程序计算在声速梯度下水中声线的传播并计算声线的长度(Program calculates the velocity gradient in the water line of acoustic sound propagation and calculate the length of the line)

共轭梯度法（Conjugate Gradient）是介于最速下降法与牛顿法之间的一个方法，它仅需利用一阶导数信息，但克服了最速下降法收敛慢的缺点，又避免了牛顿法需要存储和计算Hesse矩阵并求逆的缺点，共轭梯度法不仅是解决大型线性方程组最有用的方法之一，也是解大型非线性最优化最有效的算法之一(Conjugate gradient method (Conjugate Gradient) is between the steepest descent method and Newton' s method between a method that takes only a first derivative information, but to overcome the slow convergence of the steepest descent method shortcomings, but also avoid the need to store Newton and computing the inverse Hesse matrix and disadvantages, conjugate gradient method is not only to solve large linear equations of the most useful methods, large-scale nonlinear optimization solution is the most efficient algorithms)

这些代码可以用于产生一系列随机数，可以根据需要使用。(These codes can be used to generate a series of random numbers may be used as needed.)

计算低速气动力的涡格法程序，来自NASA，可信度高(This is a vortex lattice aerodynamic program from the low speed aerodynamics group at NASA Langley)

用C++编写的一维和二维快速傅里叶变换程序(Prepared with C++ one-dimensional and two-dimensional fast Fourier transform procedure)

说明：  数值分析实验1.明确插值多项式和分段插值多项式各自的优缺点； 编程实现拉格朗日插值算法，分析实验结果体会高次插值产生的龙格现象；(Experiments of numerical analysis 1. Define the advantages and disadvantages of interpolation polynomials and piecewise interpolation polynomials. The Lagrange interpolation algorithm is programmed, and the Runge phenomenon produced by high-order interpolation is analyzed.)

说明：  本書介紹現代的數值近似方法，解釋這些方法如何、為何，以及何時有用，以作為進一步研讀數值分析與科學計算的基礎。修習本書，學生將學會辨別須使用數值方法求解之問題的種類，及使用這些方法時，誤差傳遞的實例；對無法求得確解的問題，應能求得精確的近似解，並學會估計此近似解誤差界限的方法。書中介紹的數值方法，由數學分析、計算法則到實例，皆有系統的說明。本書對於任何要使用數值方法的學系，均可作為良好的基礎教材，同時也是一本內容完備的參考書籍。(This book introduces modern numerical approximations, explaining how, why, and when they are useful as a basis for further study of readings and scientific calculations. In this book, students will learn to identify the types of problems that must be solved using numerical methods, and examples of error propagation when using these methods; for problems that cannot be solved, an accurate approximate solution should be obtained and learn to estimate this. The numerical methods introduced in the method of approximating the error bounds are systematically explained by mathematical analysis, calculation rules and examples. This book can be used as a good basic textbook for any department that wants to use numerical methods. It is also a complete reference book.)

这是求多柔体转子纽马克法的程序。以拉格朗日方程为基础，结合有限元发计算(is seeking more flexible rotor Newmark Act procedures. To Lagrange equation, the FEM calculation)

说明：  线性代数同济第六版,最适用的一版,有书签(Linear algebra tongji sixth edition)

thermal damage abaqus-umat subroutine