Mixed Precision Iterative Refinement Techniques for the Solution of Dense Linear Systems
Aspen Plus论文
郑州大学化工与能源学院课程论文题目名称：圆锥立式混合器数值模拟专业：化学工程与工艺姓名：学号：指导老师：课程名称：工程计算方法及应用软件圆锥立式混合器数值模拟摘要：本文主要介绍应用Fluent软件对某一圆锥立式混合器的温度场的模拟，模拟中采用k —ε两方程模型，流体介质选用水，给定进口的速度，最后得出结论冷水进口速度增大，温度场分布不均性性增强，出口温度降低。
介绍了基本模拟步骤。
关键字：Fluent圆锥立式混合器温度分布进口速度模拟步骤Cone vertical mixer numerical simulationAbstract: this paper mainly introduces application of Fluent software in a cone of vertical mixer temperature field simulation, simulation of the k - epsilon two equation model, the fluid medium choose water, given the speed of import, conclusion the cold water inlet velocity increases, the temperature field distribution of sex, outlet temperature decreased. This paper introduces the basic simulation steps.Key word：Fluent cone vertical temperature distribution in the mixer imported speed simulation steps一、引言混合是把任何状态（固态、气态、液态和半液态）下的物料均匀地掺和在一起的操作，混合就是减少组分非均匀性的过程，粉体的混合有多种运动形式，以对流混合为主，其粒子由一个空间位置向另一个空间位置的运动，或两种以上组分在相互占有的空间内发生运动，以期达到各组分的均匀，混合设备是利用各种混合装置的不同组构，使粉体物料之间产生相对运动，不断改变其相对位置，并且不断克服由于物体差异导致物料分层的趋势。
udec官方讲座(english)
UDEC also simulates the behavior of the intact material between the planes of weakness as a nonlinear continuum, using the generalized finite-difference method (arbitrary element shapes), known as the finite volume method.
Discontinuous medium modeled as an assemblage or concave blocks; blocks may be rigid or deformable. Discontinuities treated as boundary conditions between blocks.
A discrete element code will embody an efficient algorithm for detecting and classifying contacts. It will maintain a data structure and memory allocation scheme that can handle many hundreds or thousands of discontinuities or contacts. The name “Distinct Element Method” is used for a DEM that uses an explicit dynamic solution to Newton’s laws of motion.
创意征集类英语作文范文
创意征集类英语作文范文Unleashing the Power of Creativity: A Journey of Innovative ThinkingIn today's rapidly evolving world, the ability to think creatively and generate innovative ideas has become a highly sought-after skill. Whether you are an entrepreneur seeking to disrupt the market, a student aiming to stand out in a competitive academic landscape, or a professional looking to drive positive change within your organization, the capacity to think outside the box and devise creative solutions is paramount. This essay delves into the art of creative idea solicitation, exploring strategies and techniques that can help you unlock your full innovative potential.The Foundation of Creativity: Cultivating a Mindset of ExplorationAt the heart of creative thinking lies a deep-rooted curiosity and a willingness to challenge the status quo. Successful innovators are not content with simply accepting the world as it is; they are driven by a relentless desire to explore new possibilities and uncover novel solutions to pressing problems. By embracing a mindset of exploration, you can begin to break free from the constraints ofconventional thinking and open yourself up to a world of untapped potential.One of the key elements in cultivating this mindset is the ability to ask the right questions. Instead of simply accepting the problem as it is presented, challenge yourself to dig deeper and uncover the underlying issues or unmet needs that may be driving the challenge. Engage in a process of active questioning, probing for alternative perspectives and unconventional approaches. This process of inquiry not only helps you gain a more comprehensive understanding of the problem but also stimulates your creative juices, as you are forced to think beyond the obvious solutions.Moreover, it is crucial to embrace a spirit of experimentation and a willingness to take calculated risks. Creativity often thrives in an environment where failure is not only accepted but also celebrated as an opportunity for learning and growth. By adopting a mindset that celebrates the journey of exploration, rather than solely focusing on the end result, you can unleash your creative potential and foster a culture of innovation within your personal or professional sphere.Techniques for Idea Generation: Unlocking the Creative Potential Once you have established a solid foundation of creative thinking, the next step is to explore a range of techniques and strategies that can help you generate a diverse array of innovative ideas. One suchapproach is the use of brainstorming sessions, where you can collaborate with a group of individuals to collectively generate a wide range of ideas, building upon each other's contributions and challenging preconceived notions.During these brainstorming sessions, it is essential to create an environment that encourages open and judgment-free discussion. Encourage participants to think freely, without fear of criticism or dismissal. Embrace the concept of "yes, and" rather than "yes, but," as this mindset fosters a spirit of collaboration and encourages the expansion of ideas, rather than their immediate rejection.Another powerful technique for idea generation is the use of mind mapping. By visually representing your thoughts, connections, and associations, you can uncover unexpected relationships and spark new avenues of exploration. Start with a central concept or problem, and then branch out, allowing your mind to make unexpected connections and generate a diverse range of ideas.Additionally, consider incorporating the use of analogies and metaphors into your creative process. By drawing parallels between seemingly unrelated domains, you can unlock new perspectives and generate innovative solutions. For example, if you are tasked with designing a more efficient transportation system, you might draw inspiration from the natural world, exploring how flocks of birds orschools of fish navigate their environments.Embracing Diversity and Collaboration: The Synergy of Collective CreativityWhile individual creativity is undoubtedly valuable, the true power of innovation often lies in the synergy of diverse perspectives and collaborative efforts. By actively seeking out and incorporating the ideas and insights of individuals from different backgrounds, disciplines, and experiences, you can unlock a wealth of creative potential that would be difficult to achieve in isolation.One effective approach to fostering this collaborative creativity is the establishment of cross-functional teams. By bringing together individuals with varying skill sets, expertise, and mindsets, you can create a dynamic environment where diverse ideas can collide and new solutions can emerge. Encourage team members to challenge each other's assumptions, share their unique perspectives, and build upon one another's contributions.Moreover, it is essential to create a culture that values and celebrates diversity. When individuals feel safe to express their unique ideas and perspectives, without fear of judgment or dismissal, the creative process can thrive. Actively seek out and embrace diverse viewpoints, as they can often challenge your own assumptions and open up new avenues of exploration.Beyond the confines of your immediate team or organization, consider reaching out to external stakeholders, such as customers, industry experts, or even individuals from completely different fields. By tapping into a broader network of perspectives, you can gain valuable insights and inspiration that may have been overlooked within your own bubble.Turning Ideas into Action: The Importance of Execution and Iteration While the generation of creative ideas is a crucial first step, the true test of innovation lies in the ability to transform those ideas into tangible and impactful solutions. This process of execution and iteration is where many promising ideas often falter, as individuals and organizations struggle to bridge the gap between conceptualization and implementation.To ensure the successful execution of your creative ideas, it is essential to develop a clear and actionable plan. Break down your vision into manageable steps, identify the resources and support required, and establish a timeline for implementation. Continuously monitor your progress, adapt to changing circumstances, and be willing to make adjustments as needed.Equally important is the ability to embrace an iterative mindset. Recognize that the first iteration of your solution may not be perfect,and be open to feedback, experimentation, and refinement. Seek out opportunities for testing and prototyping, gathering valuable insights from end-users or stakeholders, and incorporating their feedback to improve your solution.By maintaining a focus on execution and iteration, you can transform your creative ideas into tangible and impactful solutions that have the power to drive real change, whether in your personal life, your professional endeavors, or the broader societal landscape.Conclusion: Embracing the Journey of Creative Idea SolicitationThe process of creative idea solicitation is not a one-time event but rather a continuous journey of exploration, collaboration, and execution. By cultivating a mindset of curiosity, embracing diversity, and committing to the iterative process of turning ideas into action, you can unlock your full innovative potential and contribute to the advancement of your personal, professional, or societal goals.Remember, the true value of creative idea solicitation lies not only in the ideas themselves but also in the personal growth, the collaborative synergies, and the transformative impact that these ideas can have. Embrace the challenges, celebrate the failures, and continuously strive to push the boundaries of what is possible. It is through this journey of creative exploration that you can make a meaningful and lasting difference in the world around you.。
等几何分析法的程序设计流程
等几何分析法的程序设计流程The process of program design using geometric analysis methods involves several steps that help in formulating a solution to a given problem. This approach combines the principles of geometry with computational thinking to develop algorithms that solve geometric problems. In this response, I will discuss the various stages involved in the program design process using geometric analysis, highlighting the importance of each step and providing insights from different perspectives.The first step in program design using geometric analysis is problem understanding and analysis. This stage involves carefully examining the problem statement and understanding the geometric aspects involved. It is crucial to identify the key geometric properties, relationships, and constraints that need to be considered in the solution. By gaining a deep understanding of the problem, programmers can effectively develop algorithms that utilize geometric concepts to achieve the desired outcome.Once the problem is well-understood, the next step is algorithm design. This involves devising a plan or a set of instructions that outline how the problem will be solved. Geometric analysis methods often require breaking down complex problems into simpler subproblems. Algorithms can be designed to handle each subproblem individually and then combine the results to obtain the final solution. This step requires creativity and logical thinking to come up with efficient and accurate algorithms.After designing the algorithms, the next stage is implementation. This involves translating the algorithmic plan into actual code using a programming language. The code should accurately reflect the logical steps outlined in the algorithm design phase. It is important to choose an appropriate programming language and utilize relevant libraries or frameworks that support geometric operations. Additionally, the code should be well-structured, modular, and easy to understand for future maintenance and enhancements.Testing and debugging form the subsequent phase in the program design process. This step involves running the program with various test cases to ensure its correctness and efficiency. Geometric analysis methods often involve numerical computations, which can introduce errors due to rounding or approximation. Thorough testing helps identify and fix such issues. Debugging is an iterative process that involves locating and eliminating errors in the code. It requires attention to detail and a systematic approach to identify and resolve any issues that arise during testing.Documentation is an essential aspect of program design using geometric analysis. This step involves creating clear and concise documentation that describes the problem, the algorithm design, the implementation details, and any assumptions made. Documentation helps in understanding the program's functionality and aids in future maintenance or modifications. It also facilitates collaboration among team members and enables knowledge sharing.Finally, the last step in the program design process is optimization and refinement. This step focuses on improvingthe efficiency and performance of the program. Geometric analysis methods often involve complex calculations, and optimizing the code can significantly enhance its speed and resource usage. Techniques such as algorithmic improvements, data structure optimizations, and parallelization can be employed to refine the program's performance. It is important to strike a balance between optimization and maintaining code readability and maintainability.In conclusion, the program design process using geometric analysis encompasses several stages, including problem understanding, algorithm design, implementation, testing and debugging, documentation, and optimization.Each step plays a crucial role in developing a robust and efficient solution. By following this process, programmers can leverage geometric concepts and computational thinkingto solve complex problems in various domains, such as computer graphics, computer-aided design, and robotics.。
求解非线性极大极小问题的一种新的混合算法
第 ３ 卷 第４ ４ 期 ２１年 ｌ月 ０１ ２
长春理工大学学报 （ 自然科学版 ）
Ｊ ｕ ｎ ｌ ｆ ａ ｇ ｈ ｎＵｎ ｖ ｒｉ ｆＳ ｉｎ ｅａ ｄＴ ｃ ｎ ｌ ｇ （ ｔｒ ｌ ｃｅ ｃ ｉ ｏ ｏ ｒ ａ Ｃｈ ｎ ｃ ｕ ｉｅ ｓ ｙ ｏ ｃｅ ｃ ｎ ｅ ｈ ｏｏ ｙ Ｎａｕ ａ ｉｎ ｅＥｄｔ ｎ） ｏ ｔ Ｓ ｉ
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ＬＩ Ｇｕ ｚ ｉ Ｕ ｏ ｈ
（ ｏｌｇ ｆ ｃｅｃ ，Ｌａ ｎ ｇＵｎｖｒｉ ｆ ｅｒｌ ｍ ＆ Ｃ ｅ ｃｌ ｃｎ ｌｇ ．Ｆ ｓｕ １０ １ Ｃ ｌ ｅｏ ｉ ｅ ｉ ｉ ｉ ｓｙｏ Ｐ ｔ ｅ ｅ Ｓ ｎ ｏｎ ｅ ｔ ｏｕ ｈ ｍｉ ｈ ｏｏｙ ａ Ｔｅ ｕｈ ｎ１ ３ ０ ）
行基规则 ， 避免了惩罚函数法的缺点 ， 且计算结果表 ｏｅ Ｊｅｅ 方 令 ｎ表示 搜 索 空 间的 维数 ，ｚ＝（ｆｚ２… ， 明 了 Ｈｏｋ－ ｅｖｓ 法 的快 速 收 敛 性 和微 粒 群 算 － ｚ１ ｆ ｊ ｊ ｌ） ｚ 表示微粒 ｉ 当前 的位置 ， ｆＰ２… ，加 表示 微 法 的可 靠性 都得 到 了进一 步 的改善 。 Ｐ ＝（ １ ｆ ＇ ＇ Ｐ） 混 合算 法 ： 粒 ｉ 曾经 达 到 的最 好 位 置 。种群 中最优 微 粒 的序 号 第１ ： 步 随机初始化一群微粒的位置和速度 ； 用 ｇ表 示 ， 粒 ｉ 速 度 用 ：（小 ， ，ｉ） 微 的 … ＂ 表 Ｕ Ｄ 第２ ： 步 以 为 评 价 函数 ， 别对 每 个微 粒 求 问 分 示 。每个微粒根据（ ） ２ 式来更新 自己的速度和位置 ：
求解线性方程组稀疏解的稀疏贪婪随机Kaczmarz算法
大小 k̂ 。②输出 xj。③初始化 S = {1，…，n}，x0 = 0，
j = 0。④当 j ≤ M 时，置 j = j + 1。⑤选择行向量
ai，i ∈
{
1，…，n
}，每一行对应的概率为
‖a‖i
2 2
‖A‖
2 F
。
⑥
( | ) 确 定 估 计 的 支 持 集 S，S = supp xj-1 max { k̂，n-j+1 } 。
行从而达到加快算法收敛速度的目的。算法 3 给出
稀疏贪婪随机 Kaczmarz 算法。
算法 3 稀疏贪婪随机 Kaczmarz 算法。①输入
A∈ Rm×n，b ∈ Rm，最大迭代数 M 和估计的支持集的
大 小 k̂ 。 ② 输 出 xk。 ③ 初 始 化 S = {1，…，n}，x0 =
x
* 0
=
0。④
置
k
=
0
时，当
k
≤
M
-
1
时。⑤计算
( {| | } ϵk=
1 2
‖b
1 - Ax‖k 22
max
1≤ ik ≤ m
bik - aik xk 2
‖a
‖ ik
2 2
+
)1
‖A‖
2 F
（2）
⑥决定正整数指标集
{ | | } Uk =
ik|
bik - aik xk
2
≥
ϵ‖k b
-
Ax‖k
‖22 a
‖ ik
2 2
ï í
1
ï î
j
l∈S l ∈ Sc
其中，j 为迭代步数。当 j → ∞ 时，wj⊙ai → aiS，因此
QR分解算法
高等数值算法与应用高等数值算法与应（三）Advanced NumericalAdvanced Numericalg ppAlgorithms & Applications 计算机科学与技术系喻文健Cholesky factorizationSolution of symmetric indefinite systemSolution of symmetric indefinite system内容概要矩阵分解及其应用六大分解简介矩阵与线性方程组求解基本理论LU分解, Cholesky分解及其应用Q分解与线性最小乘问题QR分解与线性最小二乘问题特征值分解、奇异值分解(SVD)及其应用 稀疏矩阵的直接解法QR Factorization and Orthogonal TransformationsT f i本节主要内容QR分解方法如何做QR分解：Householder变换Givens旋转Gram-Schmidt正交化 小结QR Factorization QR FactorizationTo computer QR factorization of matrix ,A QR=To computer QR factorization of m x n matrix A , with m ≥ n , annihilate subdiagonal entries of successive columns of A , eventually reaching t i l f＝＝upper triangular form Similar to LU factorization by Gaussian正交矩阵：1.由单位正交基向量组成(2-范数,Similar to LU factorization by Gaussian elimination, but uses orthogonal transformations instead of elementary elimination matricesT⋅=Q Q I内积范数意义下)2. Possible methods include 类似作LU 分解，•Householder transformations Givens rotationsQ A R=T要消去对角线下面的元素，但采用正交变换•Givens rotations •Gram-Schmidt orthogonalization初等阵Givens QR Factorization Givens QR Factorization一般的mxm Givens 旋转矩阵，例如m=5:Aa a c s a a a ⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥112100000001αa s c a a ⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥−⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎢⎥⎣⎦334550000000000001α1旋转后的效果是将向量的第j 个分量的值“添加”到第i 个分量中。
CEM_5_solver_14
Lower triangular matrix
0 A 21 [L] = A31 AN 1
0 AN 2 AN ( N −1)
Main diagonal matrix
[ D] = diag { A11 , A22 , , ANN } N×N
l= 1, 2, , L
= (1 − ω ){x}
{x}(l +1) {x}
(l +1)
(l )
+ ω {x}
(l +1)
Algebraic field vector at the iteration step l / : Algebraic field vector at the iteration step Algebrischer Feldvektor zum Iterationsschritt l Gauss-Seidel value at the iteration step l /
0 A12 0 0 0 0 [U] = 0 0
N×N
Upper triangular matrix
A1N A( N − 2) N 0 A( N −1) N 0 0 N×N
CEM Solvers -12
Larissa Vietzorreck
The unknown xi are obtained by backsubstitution as follows:
Larissa Vietzorreck
CEM Solvers -10
4.3.1 Solving matrix equationsiterative method Jacobi´s method (component like)
变步长求积算法 英文
变步长求积算法英文Title: The Variable Step-Size Integration AlgorithmIntegration is an indispensable tool in the realm of mathematics, serving as the cornerstone for modeling and solving problems in various scientific disciplines. Among the myriad methods developed to approximate integrals, the variable step-size integration algorithm emerges as a pivotal technique, renowned for its precision and adaptability.The variable step-size integration algorithm represents a sophisticated evolution in numerical integration methods. It dynamically adjusts the size of steps based on predefined criteria, thereby enhancing accuracy where necessary while optimizing computational efficiency. This method finds extensive applications in fields as diverse as engineering, physics, economics, and beyond, where precisequantitative analysis is imperative.The allure of the variable step-size integration algorithm lies in its core principle: the adaptation of step sizes according to the function's behavior. Unlike traditional fixed-step algorithms that use a uniform step size, potentially leading to impractical computation times or inaccurate results, the variable step-size algorithm tailors its approach based on thefunction's contour. In regions where the function exhibits subtle changes, broader steps are taken, whereas zones of rapid change necessitate smaller, more precise steps. This dynamic adjustment ensures that computational resources are judiciously utilized, concentrating effort where it matters most.The implementation of the variable step-size integration algorithm typically involves a recursive or iterative process. Initially, a coarse approximation is made using large steps. As the algorithm progresses, it identifies intervals where the function varies significantly and refines the integration by reducing the step size. This process continues iteratively until the desired level of accuracy is achieved or a predetermined step size limit is reached. Through this adaptive refinement, the algorithm strikes a balance between computational cost and result precision.One quintessential example of the variable step-size algorithm is Simpson's rule with adaptive step sizes. Simpson's rule itself offers a notable improvement over basic numerical integration techniques due to its error minimization properties. When combined with a variable step-size strategy, Simpson's rule becomes even more potent. Initially, a broad estimate isobtained with large segments. The algorithm then divides the intervals with substantial variations into smaller sub-intervals, applying Simpson's rule with enhanced granularity to these specific sections. This adaptive application ensures high accuracy without needless computational expense.Another prominent method within the variable step-size integration family is Romberg integration. Romberg integration starts with the trapezoidal rule applied with wide steps and progressively halve the step size, retaining and reusing previous calculations to enhance precision. While not inherently variable in the step size, the concept can be extended to focus more computational power on intricate sections of the function by dynamically adjusting the step pattern based on function analysis.The variable step-size integration algorithm is underpinned by robust mathematical principles, ensuring both reliability and rigor. The algorithm often engages error estimation techniques, such as comparing results from successive iterations or employing different integration rules to cross-verify accuracy. These error analyses are pivotal in gauging the convergence of the algorithm and deciding whether further refinement is necessary.Error estimation in variable step integration commonly involves estimating local truncation errors and using these estimates to adjust step sizes. For instance, inRomberg integration, the Richardson extrapolation method is employed to enhance accuracy by systematically reducing errors through extrapolation rather than merely refining step sizes. This extra layer of sophistication ensures that the algorithm does not just blindly reduce step sizes but does so based on an informed error analysis.Beyond its mathematical elegance, the variable step-size integration algorithm holds immense practical significance across multiple domains. In engineering, it is instrumental in structural analysis, fluid dynamics, and thermal conduction problems where precise simulations are crucial for design and safety. In economics, it aids in the forecasting models and risk assessments where data volatility requires adaptive analytical tools. Even in modern technology, such as machine learning and data analytics, variable step-size algorithms provide robust solutions for integrating complex and dynamic systems.In the ever-evolving landscape of scientific research and technological advancement, the variable step-size integrationalgorithm stands out as a testament to human ingenuity. It encapsulates the essence of flexibility and precision, offering a nuanced approach to tackling intricate problems. As we continue to push the boundaries of knowledge, such algorithms will undoubtedly play a vital role in advancing our understanding and capabilities across diverse fields.In conclusion, the variable step-size integration algorithm exemplifies the harmonious blend of mathematical sophistication and practical utility. Its ability to fine-tune integration processes based on real-time analysis makes it an indispensable tool for scientists and engineers alike. As research progresses and computational technologies evolve, we can anticipate even more innovative enhancements to this foundational algorithm, further solidifying its status as a cornerstone of numerical analysis.。