Mosel a flexible toolset for Monadic Second-order Logic
VHDL5套试卷标准答案
填空题：1、一般将一个完整的VHDL程序称为设计实体2、VHDL设计实体的基本结构由（库）、（程序包）、（实体）、（结构体）和（配置）组成。
3、（实体）和（结构体）是设计实体的基本组成部分，它们可以构成最基本的VHDL 程序。
4、根据VHDL语法规则，在VHDL程序中使用的文字、数据对象、数据类型都需要（事先声明）。
5、在VHDL中最常用的库是（IEEE）标准库，最常用的数据包是（STD_LOGIC_1164）数据包。
6、VHDL的实体由（实体声明）部分和（结构体）组成。
7、VHDL的实体声明部分指定了设计单元的（输入出端口）或（引脚），它是设计实体对外的一个通信界面，是外界可以看到的部分。
8、VHDL的结构体用来描述实体的（逻辑结构）和（逻辑功能），它由VHDL 语句构成，是外界看不到的部分。
9、在VHDL的端口声明语句中，端口方向包括（输入）、（输出）、（双向）和（缓冲）。
10、VHDL的标识符名必须以（字母开头），后跟若干字母、数字或单个下划线构成，但最后不能为（下划线）11、VHDL的数据对象包括（常量）、（变量）和（信号），它们是用来存放各种类型数据的容器。
12、为信号赋初值的符号是（：=）；程序中，为变量赋值的符号是（：=），为信号赋值的符号是（<=）13、VHDL的数据类型包括（标量类型）、（复合类型）、（存储类型）和（文件类型）。
14、在VHDL中，标准逻辑位数据有（九）种逻辑值。
15、VHDL的操作符包括（逻辑）、（算术）、（关系）和（并置）四类。
选择题：1、IEEE于1987年公布了VHDL的（A）语法标准。
A、IEEE STD 1076-1987;B、RS232;C、IEEE STD_LOGIC_1164;D、IEEE STD 1076-1993;2、IEEE于1987年公布了VHDL的（D）语法标准。
A、IEEE STD 1076-1987;B、RS232;C、IEEE STD_LOGIC_1164;D、IEEE STD 1076-1993;3、VHDL的设计实体可以被高层次的系统（D ），成为系统的一部分。
Maple教程
Maple 软件使用教程目录序言 0第一章Maple概述 (2)第二章基本命令 (4)第三章作图 (11)§3.1 .二维曲线图 (13)§3.2三维图形 (16)§3.3数据图 (17)实验一第四章微积分 (20)§4.1函数 (18)§4.2极限 (19)§4.3导数 (20)§4.4积分 (21)§4.5方程求解 (22)§4.6极值与最值 (23)§4.7微分方程与差分方程 (24)§4.8级数 (25)实验二第五章线性代数 (32)实验三第六章概率统计 (35)§6.1描述性数据分析discribe (28)§6.2拟合回归分析 (29)§6.3数据形式变换transform (30)§6.4按分布产生随机数random (30)§6.5分布的数据计算statevalf (31)§6.6统计绘图statplots (32)§6.7方差分析anova (32)第七章线性规划.................... ............................... .. (40)第八章程序语句 (41)实验四附录一 ..................................................................................................... .. (43)附录二...…...…………..……..……………………………………………. .43.第一章Maple概述Maple以其良好的使用环境、超强的符号计算、高精度的数值计算、方便的图形处理和简洁而高效的编程功能，越来越受到大家的喜爱和重视。
为了让同学们了解什么是Maple以及Maple能解决什么问题，我们先来介绍Maple的初步知识、基本功能及简单的历史发展。
Verilog_HDL复杂数字系统设计-2_[1]...
![Verilog_HDL复杂数字系统设计-2_[1]...](https://img.taocdn.com/s3/m/3f4f2f7ba26925c52cc5bff2.png)
2013-8-4
南通大学电子信息学院
7
1.3复杂数字系统的设计方法
1.3.1 复杂数字逻辑系统
• 嵌入式微处理机系统
• 数字信号处理系统 • 高速并行计算逻辑
• 高速通信协议电路
• 高速编码/解码、加密/解密电路 • 复杂的多功能智能接口
• 门逻辑总数超过几万门达到几百甚至达几千万门的数
字系统
2013-8-4 南通大学电子信息学院 8
1990 Verilog HDL 公开发表
1995 Verilog IEEE1364 标准公开发表
1990有关Verilog HDL的 全部权利都移交给OVI(Open Verilog International)组织
2013-8-4 南通大学电子信息学院 6
1.2.3 Verilog HDL的优点
2013-8-4
南通大学电子信息学院
24
例2-2b 4选1多路选择器
module mux4_to_1 (out, i0, i1, i2, i3, s1, s0); output out; input i0, i1, i2, i3; input s1, s0; wire s1n, s0n; wire y0, y1, y2, y3; not not0(s1n, s1); not not1(s0n, s0); and and0(y0, i0, s1n, s0n); and and1(y1, i1, s1n, s0); and and2(y2, i2, s1, s0n); and and3(y3, i3, s1, s0); or or0(out, y0, y1, y2, y3); endmodule
2013-8-4 南通大学电子信息学院 3
Selfish routing in capacitated networks
Date : June 2003; revised February 2004.
2000 Mathematics Subject Classification. Primary 90C35; 90B10, 90B20, 90C25, 90C27, 91A10, 91A13,
91A43.
Key words and phrases. Selfish Routing, Price of Anarchy, Traffic Assignment, System Optimum, Nash
SELFISH ROUTING IN CAPACITATED NETWORKS
´ R. CORREA, ANDREAS S. SCHULZ, AND NICOLAS ´ E. STIER MOSES JOSE Sloan School of Management and Operations Research Center Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139-4307
Abstract. According to Wardrop’s first principle, agents in a congested network choose their routes selfishly, a behavior that is captured by the Nash equilibrium of the underlying noncooperative game. A Nash equilibrium does not optimize any global criterion per se, and so there is no apparent reason why it should be close to a solution of minimal total travel time, i.e. the system optimum. In this paper, we offer positive results on the efficiency of Nash equilibria in traffic networks. In contrast to prior work, we present results for networks with capacities and for latency functions that are nonconvex, nondifferentiable, and even discontinuous. The inclusion of upper bounds on arc flows has early been recognized as an important means to provide a more accurate description of traffic flows. In this more general model, multiple Nash equilibria may exist and an arbitrary equilibrium does not need to be nearly efficient. Nonetheless, our main result shows that the best equilibrium is as efficient as in the model without capacities. Moreover, this holds true for broader classes of travel cost functions than considered hitherto.
焊接温度场和应力场的数值模拟
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沈阳工业大学硕士学位论文焊接温度场和应力场的数值模拟姓名：王长利申请学位级别：硕士专业：材料加工工程指导教师：董晓强 20050310沈阳工业大学硕士学位论文摘要焊接是一个涉及电弧物理、传热、冶金和力学的复杂过程。
焊接现象包括焊接时的电磁、传热过程、金属的熔化和凝固、冷却时的相变、焊接应力和变形等。
一旦能够实现对各种焊接现象的计算机模拟，我们就可以通过计算机系统来确定焊接各种结构和材料的最佳设计、最佳工艺方法和焊接参数。
本文在总结前人的工作基础上系统地论述了焊接过程的有限元分析理论，并结合数值计算的方法，对焊接过程产生的温度场、应力场进行了实时动态模拟研究，提出了基于ＡＮＳＹＳ软件为平台的焊接温度场和应力场的模拟分析方法，并针对平板堆焊问题进行了实例计算，而且计算结果与传统结果和理论值相吻合。
本文研究的主要内容包括：在计算过程中材料性能随温度变化而变化，属于材料非线性问题；选用高斯函数分布的热源模型，利用函数功能实现热源的移动。
建立了焊接瞬态温度分布数学模型，解决了焊接热源移动的数学模拟问题；通过改变单元属性的方法，解决材料的熔化、凝固问题；对焊缝金属的熔化和凝固进行了有效模拟，解决了进行热应力计算收敛困难或不收敛的问题；对焊接过程产生的应力进行了实时动态模拟，利用本文模拟分析方法，可以对焊接过程的热应力及残余应力进行预测。
本文建立了可行的三维焊接温度场、应力场的动态模拟分析方法，为优化焊接结构工艺和焊接规范参数，提供了理论依据和指导。
关键词：焊接，数值模拟，有限元，温度场，应力场沈阳工业大学硕士学位论文ＳｉｍｕｌａｔｉｏｎｏｆｗｅｌｄｉｎｇｔｅｍｐｅｒａｔｕｒｅｆｉｅｌｄａｎｄｓｔｒｅｓｓｆｉｅｌｄＡｂｓｔｒａｃｔＷｅｌｄｉｎｇｉｓａｃｏｍｐｌｉｃａｔｅｄｐｈｙｓｉｃｏｃｈｅｍｉｃａ／ｐｒｏｃｅｓｓｗｌｆｉｅｈｉｎｖｏｌｖｅｓｉｎｅｌｅｃｔｒｏｍａｇｎｅｔｉｓｍ，Ｍａｔｔｒａｎｓｆｅｒｒｉｎｇ，ｍｅｔａｌｍｅｌｔｉｎｇａｎｄｆｒｅｅｚｉｎｇ，ｐｈａｓｅ?ｃｈａｎｇｅｗｅｌｄｉｎｇＳＯｓｔｒｅｓｓａｎｄｄｅｆｏｒｍａｔｉｏｎａｎｄｏｎ，Ｉｎｏｒｄｅｒｔｏｇｅｔｈｉｇｈｑｕａｆｉｔｙｗｅｌｄｉｎｇｓｔｍｃｔｔｌｒｅ，ｔｈｅｓｅｆａｃｔｏｒｓｈａｖｅｔｏｂｅｃｏｎｔｒｏｌｌｅｄ．Ｉｆｃａｎｗｅｌｄｉｎｇｐｒｏｃｅｓｓｂｅｓｉｍｕｌａｔｅｄｗｉｔｈｃｏｍｐｕｔｅｒ，ｔｈｅｂｅｓｔｄｅｓｉｇｎ，ｐｍｃｅｄｕｒｅｍｅｔｈｏｄａｎｄｏｐｔｉｍｕｍｗｅｌｄｉｎｇｐａｒａｍｅｔｅｒｃａｎｂｅｏｂｔａｉｎｅｄ．ＢａｓｅｄＯｉｌｓｕｍｍｉｎｇｕｐｏｔｈｅｒ’Ｓｅｘｐｅｒｉｅｎｃｅ，ｅｍｐｌｏｙｉｎｇｎｕｍｅｒｉｃａｌｃａｌｃｕｌａｔｉｏｎｍｅｔｈｏｄ，ｔｈｉｓｐａｐｅｒｒｅｓｅａｒｃｈｅｒｓｙｓｔｅｍｉｃａｌｌｙｄｉｓｃｕｓｓｅｓｔｈｅｆｉｎｉｔｅｅｌｅｍｅｎｔａｎａｌ删ｓｙｓｔｅｍｏｆｔｈｅｗｅｌｄｉｎｇｐｒｏｃｅｓｓｂｙｒｅａｌｉｚｉｎｇｔｈｅ３Ｄｄｙｎａｍｉｃｓｉｍｕｌａｔｉｏｎｏｆｗｅｌｄｉｎｇｔｅｍｐｅｒａｔｕｒｅｆｉｅｌｄａｎｄｓｔｒｅｓｓｆｉｅｌｄ，ｔｈｅｎｕｓｅｓｔｈｅｒｅｓｅａｒｃｈｒｅｓｕｌｔｔｏｓｉｍｕｌａｔｅｔｈｅｗｅｌｄｉｎｇｐｒｏｃｅｓｓｏｆｂｏａｒｄｓｕｒｆａｃｉｎｇｂｙＦＥＭｓｏｆｔＡＮＳＹＳ．Ａｔｔｈｅｔｈｅｏｒｙｒｅｓｕｌｔ．ｓａｍｅｔｉｍｅ．ｔｈｅｃａｌｃｕｌａｔｉｏｎｒｅｓｕｌｔａｃｃｏｒｄｓｗｉｔｈｔｒａｄｉｔｉｏｎａｌａｎａｌｙｓｉｓｒｅｓｕｌｔａｎｄＴｈｅｍａｉｎｃｏｎｔｅｎｔｓｏｆｔｈｅｐａｐｅｒａｒｅａｓｆｏｌｌｏｗｉｎｇ：ｔｈｅｃａｌｃｕｌａｔｉｏｎｉｎｗｅｌｄｉｎｇｐｒｏｃｅｓｓｉｓａｍａｔｅｒｉａｌｎｏｎｌｉｎｅａｒｐｒｏｃｅｄｕｒｅｔｈａｔｔｈｅｍａｔｅｒｉａｌｐｒｏｐｅｒｔｉｅｓｃｈａｎｇｅｔｈｅｆｕｎｃｔｉｏｎｏｆＧａｕｓｓａｓｗｉｔｈｔｈｅｔｅｍｐｅｒａｔｕｒｅ；ｃｈｏｏｓｅｈｅａｔｓｏｕｒｃｅｍｏｄｅｌ．ｕｓｅｔｈｅｆｕｎｃｔｉｏｎｃｏｍｍａｎｄｔｏａｐｐｌｙｌｏａｄｏｆｍｏｖｉｎｇｈｅａｔＳ０１２Ｉｅ－２．ＡｍａｔｈｅｍａｔｉｃｍｏｄｅｌｏｆｔｒａｎｓｉｅｎｔｔｈｅｒｍａｌｐｒｏｃｅｓｓｉｎｗｅｌｄｉｎｇｉｓｅｓｔａｂｌｉｓｈｅｄｔｏｓｉｍｕｌａｔｅｔｈｅｍｏｖｉｎｇｏｆｔｈｅｈｅａｔｓｏＢｒｃｅ．Ｔｈｅｅｆｆｅｃｔｓｏｆｍｅｓｈｓｉｚｅ，ｗｅｌｄｉｎｇｓｐｅｅｄ，ｗｅｌｄｉｎｇｃｕｒｒｅｎｔａｎｄｅｆｆｅｃｔｉｖｅｒａｄｉｕｓｅｌｅｃｔｒｉｃａｒｃｏｎｔｅｍｐｅｒａｔｕｒｅｆｉｅｌｄａ比ｄｉｓｃｕｓｓｅｄ．Ｔｈｅｐｒｏｂｌｅｍｏｆｔｈｅｆｕｓｉｏｎａｎｄｓｏｌｉｄｉｆｉｃａｔｉｏｎｏｆｍａｔｅｒｉａｌｈａｓｂｅｅｎｓｏｌｖｅｄｂｙｔｈｅｍｅｔｈｏｄｏｆｃｈａｎｇｉｎｇｔｈｅｅｌｅｍｅｎｔｍａｔｅｒｉａｌ．Ｔｈｅｐｒｏｂｌｅｍｏｆｔｈｅｃｏｎｖｅｒｇｅｎｃｅｄｉｆｆｉｃｕｌｔｙｏｒｔｈｅｕｎ—ｃｏｎｖｅｒｇｅｎｃｅｄｕｒｉｎｇｔｈｅｃａｌｃｕｌａｔｉｎｇｏｆｔｈｅｔｈｅｒｍａｌｓｌＴｅｓｓｉｓｓｏｌｖｅｄ；ｔｈｒｏｕｇｈｒｅａｌ－ｔｉｍｅｄｙｎａｍｉｃｓｉｍｕｌａｔｉｏｎｏｆｔｈｅｓｔｒｅｓｓｐｒｏｄｕｃｅｄｉｎｗｅｌｄｉｎｇｐｒｏｃｅｓｓ，ｔｈｅｔｈｅｒｍａｌｓｔｒｅｓｓａｎｄｒｅｓｉｄｕａｌＳｌｌ℃ＳＳｉｎｗｅｌｄｉｎｇｃａｎｂｅｐｒｅｄｉｃｔｅｄｂｙｕｓｉｎｇｔｈｅｓｉｍｕｌａｔｉｖｅａｎａｌｙｓｉｓｍｅｔｈｏｄｉｎｔｈｉｓｐａｐｅｒ．Ｉｎｔｈｉｓｐａｐｅｒ，ａｆｅａｓｉｂｌｅｓｌＩｅｓｓｄｙｎ黜ｆｉｅｓｉｍｕｌａｔｉｏｎｍｅｔｈｏｄｏｎ３Ｄｗｅｌｄｉｎｇｔｅｍｐｅｒａｔｕｒｅｆｉｅｌｄ，ｏｎｆｉｅｌｄｈａｄｂｅｅｎｅｓｔａｂｌｉｓｈｅｄ，ｗｈｉｃｈｐｒｏｖｉｄｅｓｔｈｅｏｒｙｆｏｕｎｄａｔｉｏｎａｎｄｉｎｓｔｒｕｃｔｉｏｎｏｐｔｉｍｉｚｉｎｇｔｈｅｗｅｌｄｉｎｇｔｅｃｈｎｏｌｏｇｙａｎｄｐａｒａｍｅｔｅｒｓ．ＫＥＹＷＯＲＤ：Ｗｅｌｄｉｎｇ，ＮｕｍｅｒｉｃａｌＳｉｍｕｌａｔｉｏｎ，Ｆｉｎｉｔｅｅｌｅｍｅｎｔ，Ｔｅｍｐｅｒａｔｕｒｅｆｉｅｌｄ，Ｓｔｒｅｓｓｆｉｅｌｄ．２．独创性说明本人郑重声明：所呈交的论文是我个人在导师指导下进行的研究工作及取得的研究成果。
一个自由模态的摹状词理论LFMDTK
一个自由模态的摹状词理论LFMDT K*冯 艳 刘壮虎1、L K的理论背景用自由逻辑来处理限定摹状词，已形成多种自由摹状词理论。
这些理论以自由逻辑为基础，是自由逻辑在摹状词领域的重要应用。
但是，对自然语言中如“当今的法国国王是当今的法国国王，这是必然的”、“唐僧的大徒弟会七十二变是必然的”这样的既包含“空限定摹状词”，又包含“必然”的语句，却很难用这些理论来刻画。
本文提出一个自由模态的摹状词理论LFMDT K（文中简记为“L K”），来处理和刻画限定摹状词在模态语境中出现的情形。
在L K中，引入了模态算子“□”。
L K能够反映出限定摹状词与个体常量、个体变元的根本区别。
个体常量和个体变元是严格指示词，即对任意可能世界w∈W，个体常量或个体变元在w中的指称与它们在任意w所通达的可能世界中的指称相同。
而摹状词是非严格指示词，在不同的可能世界中可能会有不同的指称。
比如，“世界上最高的山峰”，在现实世界中指“珠穆朗玛峰”，而在可能世界w1中，可能珠穆朗玛峰存在但不是最高的山峰，而有另外一座叫做“无名峰”的山峰是最高的，这时“世界上最高的山峰”在w1中就指“无名峰”，而不指“珠穆朗玛峰”。
在w1中，也可能珠穆朗玛峰根本就不存在，在这种情况下，显然“世界上最高的山峰”也不会指“珠穆朗玛峰”。
2、L K的形式语言及其系统2.1 L K的形式语言L K的初始符号是通常的带等词的模态谓词逻辑初始符号的扩充，加上表示“存在”的逻辑谓词E!和摹状算子ι。
谓词用P，Q，R等表示，变元用x，y，z等表示，常量用a，b，c等表示。
L K的项和公式形成规则项用t, s等表示，公式用A, B等表示，项和公式必要时都可以加下标。
(1) 变元和常量是项；(2) 如果t是项，则E!t是公式；(3) 如果t，s是项，则t≡s是公式；(4) 如果t1, t2, …, t n是项，P是n元谓词，则Pt1…t n是公式；（P可以是≡或E!）(5) 如果A是公式，则¬A是公式；(6) 如果A, B是公式，则(A→B)是公式；(7) 如果A是公式，则□A是公式；(8) 如果A是公式，x是变元，则∀xA是公式；(9) 如果A是公式且其中不含□，x是变元，则ιxA是项。
自然语言处理
自然语言处理2002．11．09中国科学院计算技术研究所1.综述.1.1. 绪论.1.1.1.背景,目标.1.1.1.1. 研究自然语言的动力1．语言是思维的裁体，是人际交流的重要工具。
在人类历史上以语言文字形式记载和流传的知识占到知识总量的80％以上。
就计算机的应用而言，据统计用于数学计算的仅占10％，用于过程控制的不到5％，其余85％左右都是用于语言文字的信息处理。
在这样的社会需求下，自然语言理解作为语言信息处理技术的一个高层次的重要方向，一直是人工智能界所关注的核心课题之一。
2．由于创造和使用自然语言是人类高度智能的表现，因此对自然语言理解的研究也有助于揭开人类智能的奥秘，深化我们对语言能力和思维本质的认识。
.1.1.1.2. 什么是计算语言学计算语言学（Computational Linguistics）指的是这样一门学科，它通过建立形式化的数学模型，来分析、处理自然语言，并在计算机上用程序来实现分析和处理的过程，从而达到以机器来模拟人的部分乃至全部语言能力的目的。
计算语言学（Computational Linguistics）有时也叫计量语言学（Quantitative Linguistics）, 数理语言学（Mathematical Linguistics）, 自然语言理解（Natural Language Understanding）, 自然语言处理（Natural Language Processing）, 人类语言技术（Human Language Technology）。
.1.1.1.3. 图灵测验在人工智能界，或者语言信息处理领域中，人们普遍认为可以采用著名的1950年描述的图灵试验(Turing Test )来判断计算机是否“理解”了某种自然语言。
.1.1.1.3.1.Turing模仿游戏(Imitation Game)●场景：男性被试、女性被试、观察者，3者在3个不同的房间，房间号分别为X, Y, O●规则：观察者用电传打字机与被试们通信，男性被试欺骗观察者、女性被试帮助观察者。
数学-科学的王后与仆人
数学: 科学的王后和仆人Mathematics: Queen and Servant of Science北京理工大学叶其孝本文的题目是已故的美国科学院院士、著名数学家、数学史学家和科普作家Eric Temple Bell(贝尔, 1883, 02, 07 ~ 1960, 12, 21)于1951年写的一本书的书名Mathematics: Queen and Servant of Science (数学: 科学的王后和仆人). 该书主要是为大学生和非数学领域的人士写的, 介绍纯粹和应用数学的各个方面, 更着重在说明数学科学的极端重要性.The Mathematical Association of America, 1996, 463 pages实际上这是他1931年写的The Queen of the Sciences (科学的王后)和1937年写的The Handmaiden of the Sciences (科学的女仆)这两本通俗数学论著的合一修订扩大版.Eric Temple Bell Alexander Graham Bell (1847 ~ 1922) 按常识的理解, 女王是优美、高雅、无懈可击、至尊至贵的, 在科学中只有纯粹数学才具有这样的特点, 简洁明了的数学定理一经证明就是永恒的真理, 极其优美而且无懈可击;另一方面, 科学和工程的各个分支都在不同程度上大量应用数学, 这时数学科学就是仆人, 这些仆人是否强有力, 用起来是否得心应手是雇佣这些仆人的主人最为关心的事. 事实上, servant这个字本身就有“供人们利用之物, 有用的服务工具”的意思. 毫无疑问, 我们的目的不是为数学争一个好的名分, 而是想说明数学是怎样通过数学建模来解决各种实际问题的; 数学(数学建模)的极端重要性, 以及探讨正确认识和理解数学科学的作用对于发展我国科学技术、经济以及教育, 从而争取在21世纪把我国真正建设成为屹立于世界民族之林的强国，乃至个人事业发展的至关重要性. 当然, 我们也希望说明王后和仆人集于一身并不矛盾. 历史上, 很多特别受人尊敬的科学家, 不仅仅是由于他们的科学成就, 更因为他们的科学成就能够服务于人类.数学是科学的王后, 算术是数学的王后. 她常常放下架子为天文学和其他科学效劳, 但是在所有情况下, 第一位的是她(数学)应尽的责任. (高斯)Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics. She often condescends to render service to astronomy and other natural sciences, but under all circumstance the first place is her due.— Carl Friedrich Gauss (卡尔·弗里德里希·高斯, 1777, 4, 30 ~ 1855, 2, 23)From: Bell, Eric T., Mathematics: Queen and Servant of Science, MAA, 1951, p.1;Men of Mathematics, Simon and Schuster, New York, 1937, p. xv.***************************************************自古以来，数学的发展始终与科学技术的发展紧密相连，反之亦然. 首先, 我们来看一下导致我们现在这个飞速发展的信息社会的19、20世纪几乎所有重大科学理论的发展和完善过程中数学(数学建模)所起到的不可勿缺的作用.数学研究的成果往往是重大科学发明的催生素(仅就19、20世纪而言, 流体力学、电磁理论、相对论、量子力学、计算机、信息论、控制论、现代经济学、万维网和互联网搜索引擎、生物学、CT、甚至社会政治学领域等). 但是20世纪上半世纪, 数学虽然也直接为工程技术提供一些工具, 但基本方式是间接的: 先促进其他科学的发展, 再由这些科学提供工程原理和设计的基础. 数学是幕后的无名英雄.现在, 数学无处不在, 数学和工程技术之间,在更广阔的范围内和更深刻的程度上, 直接地相互作用着, 极大地推动了科学和工程科学的发展, 也极大地推动了技术的发展. 数学不仅是幕后的无名英雄, 很多方面开始走向“前台”. 但是对数学的极端重要性迄今尚未有共识, 取得共识对加强一个国家的竞争力来说是至关重要的.硬能力―一位美国朋友谈及对未来中国人的看法: 20年后, 中国年轻人会丢了中国人现在的硬能力, 他们崇拜各种明星, 不愿献身科学, 不再以学术研究为荣, 聪明拔尖的学生都去学金融、法律等赚钱的专业; 而美国人因为认识到其硬能力(例如数学)不行, 进行教育改革, 20年后, 不但保持了其软实力即非专业能力的优势, 而且在硬能力上赶上中国人.‖“正在丢失的硬实力”, 鲁鸣, 《青年文摘》2011年第5期动向：美国很多州新办STEM高中, 一些大学开始开设STEM课程等.STEM = Science + Technology + Engineering + Mathematics2012年2月7日公布的美国总统科技顾问委员会给总统的报告，参与超越：培养额外的100万具有科学、技术、工程和数学学位的大学生(Engage to Excel: Producing One Million Additional College Graduates with Degrees in Science, Technology, Engineering, and Mathematics)The Mathematical Sciences in 2025, the National Academies Press, 2013人们使用的数学科学思想、概念和方法的范围在不断扩大的同时，数学科学的用途也在不断扩展. 21世纪的大部分科学与工程将建立在数学科学的基础上.This major expansion in the uses of the mathematical sciences has been paralleled by a broadening in the range of mathematical science ideas and techniques being used. Much of twenty-first century science and engineering is going to be built on a mathematical science foundation, and that foundation must continue to evolve and expand.数学科学是日常生活的几乎每个方面的组成部分.互联网搜索、医疗成像、电脑动画、数值天气预报和其他计算机模拟、所有类型的数字通信、商业和军事中的优化问题以及金融风险的分析——普通公民都从支撑这些应用功能的数学科学的各种进展中获益，这样的例子不胜枚举.The mathematical sciences are part of almost every aspect of everyday life. Internet search, medical imaging, computer animation, numerical weather predictions and othercomputer simulations, digital communications of all types, optimization in business and the military, analyses of financial risks —average citizens all benefit from the mathematical science advances that underpin these capabilities, and the list goes on and on.调查发现:数学科学研究工作正日益成为生物学、医学、社会科学、商业、先进设计、气候、金融、先进材料等许多研究领域不可或缺的重要组成部分. 这种研究工作涉及最广泛意义下数学、统计学和计算综合，以及这些领域与潜在应用领域的相互作用. 所有这些活动对于经济增长、国家竞争力和国家安全都是至关重要的，而且这种事实应该对作为整体的数学科学的资助性质和资助规模产生影响. 数学科学的教育也应该反映数学科学领域的新的状况.Finding: Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and many more. This work involves the integration of mathematics, statistics, and computation in the broadest sense and the interplay of these areas withareas of potential application. All of these activities are crucial to economic growth, national competitiveness, and national security, and this fact should inform both the nature and scale of funding for the mathematical sciences as a whole. Education in the mathematical sciences should also reflect this new stature of the field.****************************************************************为了以下讲述的方便, 我们先来了解一下什么是数学建模.数学模型(Mathematical Model)是用数学符号对一类实际问题或实际发生的现象的(近似的)描述.数学建模(Mathematical Modeling)则是获得该模型并对之求解、验证并得到结论的全过程.数学建模不仅是了解基本规律, 而且从应用的观点来看更重要的是预测和控制所建模的系统的行为的强有力的工具.数学建模是数学用来解决各种实际问题的桥梁.↑→→→→→→→→↓↑↓↑↓↓↑↓←←←←←通不过↓↓通过)定义：数学建模就是上述框图多次执行的过程数学建模的难点观察、分析实际问题, 作出合理的假设, 明确变量和参数, 形成明确的数学问题. 不仅仅是翻译的问题; 涉及的数学问题可能是复杂、困难的, 求解也许涉及深刻的数学方法. 如何作出正确的判断, 寻找合适、简洁的(解析或近似) 解法; 如何验证模型.简言之:合理假设、模型建立、模型求解、解释验证.记住这16个字, 将会终生受用.数学建模的重要作用：源头创新当然数学建模也有局限性, 不能单独包打天下, 因为实际问题是非常复杂的, 需要多学科协同解决.在图灵(A. M. Turing)的文章: The Chemical Basis of Morphogenesis (形态生成的化学基础), Philosophical Transactions of the Royal Society of London (伦敦皇家学会哲学公报), Series B (Biological Sciences),v.237(1952), 37-72.1. 一个胚胎的模型. 成形素本节将描述一个正在生长的胚胎的数学模型. 该模型是一种简化和理想化, 因此是对原问题的篡改. 希望本文论述中保留的一些特征, 就现今的知识状况而言, 是那些最重要的特征.1. A model of the embryo. MorphogensIn this section a mathematical model of the growing embryo will be described. This model will be asimplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge.想单靠数学建模本身来解决重大的生物学问题是不可能的，另一方面，想仅仅依靠实验来获得对生物学的合理、完整的理解也是极不可能的. There is no way mathematical modeling can solve major biological problems on its own. On the other hand, it ishighly unlikely that even a reasonably complete understanding could come solely from experiment.—— J. D. Murray, Why Are There No 3-Headed Monsters? Mathematical Modeling in Biology, Notices of the AMS,v. 59 (2012), no. 6, p.793.自古以来公平、公正的竞赛都是培养、选拔人才的重要手段, 科学和数学也不例外.中学生IMO (国际数学奥林匹克(International Mathematical Olympiad), 1959 ～)北美的大学生Putnbam数学竞赛(1938 ～)全国大学生数学竞赛(2010 ～)Mathematical Contest in Modeling (MCM, 1985 ～)美国大学生数学建模竞赛Interdisciplinary Contest in Modeling (ICM, 1999～)美国大学生跨学科建模竞赛China Undergraduate Mathematical Contest in Modeling (CUMCM, 1992～) 中国大学生数学建模竞赛中国大学生参加美国大学生数学建模竞赛情况中国大学生数学建模竞赛情况在以下讲述中涉及物理方面的具体的数学模型 (问题)的叙述和初步讨论可参考《物理学与偏微分方程》, 李大潜、秦铁虎编著, (上册, 1997; 下册, 2000), 高等教育出版社.Seven equations that rule your world (主宰你生活的七个方程式), by Ian Stewart, NewScientist, 13 February 2012.Fourier transformation 2ˆ()()ix f f x e dx πξξ∞--∞=⎰Wave equation 22222u u c t x ∂∂=∂∂ Ma xwell‘s equation110, , 0, H E E E H H c t c t∂∂∇⋅=∇⨯=-∇⋅=∇⨯=∂∂Schrödinger‘s equation ˆψH ψi t∂=∂Ian Stewart, In Pursuit of the Unknown:17 Equations That Changed the World (追求对未知的认识：改变世界的17个方程), Basic Books, March 13, 2012.目录(Contents)Why Equations? /viii1. The squaw on the hippopotamus ——Pythagoras‘sTheorem/12. Shortening the proceedings —— Logarithms/213. Ghosts of departed quantities —— Calculus/354. The system of the world ——Newton‘s Law ofGravity/535. Portent of the ideal world —— The Square Root ofMinus One/736. Much ado about knotting ——Euler‘s Formula forPolyhedra/837. Patterns of chance —— Normal Distribution/1078. Good vibrations —— Wave Equation/1319. Ripples and blips —— Fourier Transform/14910. The ascent of humanity —— Navier-StokesEquation/16511. Wave in the ether ——Maxwell‘s Equations/17912. Law and disorder —— Second Law ofThermodynamics /19513. One thing is absolute —— Relativity/21714. Quantum weirdness —— Schrödinger Equation/24515. Codes, communications, and computers ——Information Theory/26516. The imbalance of nature —— Chaos Theory/28317. The Midas formula —— Black-Scholes Equation/195Where Next?/317Notes/321Illustration Credits/330Index/331相对论Albert Einstein(1879, 3, 14 ～1955, 4, 18)20世纪最伟大的科学成就莫过于Einstein(爱因斯坦)的狭义和广义相对论了, 但是如果没有Minkowski (闵可夫斯基)几何、Riemann(黎曼)于1854年发明的Riemann几何, 以及Cayley(凯莱), Sylvester(西勒维斯特)和Noether(诺特)等数学家发展的不变量理论, Einstein的广义相对论和引力理论就不可能有如此完善的数学表述. Einstein自己也不止一次地说过.早在1905年, 年仅26岁的爱因斯坦就已提出了狭义相对论. 狭义相对论推倒了牛顿力学的质量守恒、能量守恒、质量能量互不相关、时空永恒不变的基本命题. 这是一场真正的科学革命.为了导出狭义相对论，爱因斯坦作出了两个假设：运动的相对性(所有匀速运动都是相对的)和光速为常数(光的运动例外, 它是绝对的). (1)狭义相对性原理,即在所有惯性系中, 物理学定律具有相同的数学表达形式;(2)光速不变原理,真空中光沿各个方向传播的速率都相等，与光源和观察者的运动状态无关.时空不是绝对独立的.由此可以导出一些推论: 相对论坐标变换式和速度变换式, 同时的相对性, 钟慢尺缩效应和质能关系式等.他的好友物理学家P.Ehrenfest指出实际上还蕴涵着第三个假设, 即这两个假设是不矛盾的. 物体运动的相对性和光速的绝对性, 两者之间的相互制约和作用乃是相对论里一切我们不熟悉的时空特征的根源.(部分参阅李新洲:《寻找自然之律--- 20世纪物理学革命》, 上海科技教育出版社, 2001.)1907 年德国数学家H. Minkowski (1864 ～1909) 提出了―Minkowski 空间‖,即把时间和空间融合在一起的四维空间1,3R. Minkowski 几何为Einstein 狭义相对论提供了合适的数学模型.“没有任何客观合理的方法能够把四维连续统分离成三维空间连续统和一维时间连续统. 因此从逻辑上讲, 在四维时空连续统(space- time continuum)中表述自然定律会更令人满意. 相对论在方法上的巨大进步正是建立在这个基础之上的, 这种进步归功于闵可夫斯基(Minkowski).”—Albert Einstein, The Meaning of Relativity, 1922, Princeton University Press. 中译本, 阿尔伯特·爱因斯坦著, 相对论的意义, (普林斯顿科学文库(Princeton Science Library) 1), 郝建纲、刘道军译, 上海科技教育出版社, 2001, p. 27.有了Minkowski 时空模型后, Einstein 又进一步研究引力场理论以建立广义相对论. 1912 年夏他已经概括出新的引力理论的基本物理原理, 但是为了实现广义相对论的目标, 还必须寻求理论的数学结构, Einstein 为此花了 3 年的时间, 最后, 在数学家M. Grossmann 的介绍下学习掌握了发展相对论引力学说所必需的数学工具—以Riemann几何和Ricci, Levi - Civita的绝对微分学, 也就是Einstein 后来所称的张量分析.“根据前面的讨论, 很显然, 如果要表达广义相对论, 就需要对不变量理论以及张量理论加以推广. 这就产生了一个问题, 即要求方程的形式必须对于任意的点变换都是协变的. 在相对论产生以前很久, 数学家们就已经建立了推广的张量演算理论. 黎曼(Riemann)首先把高斯(Gauss)的思路推广到了任意维连续统, 他很有预见性地看到了……进行这种推广的物理意义. 随后, 这个理论以张量微积分的形式得到了发展, 对此里奇(Ricci)和莱维·齐维塔(Tulio Levi-Civita, 1873～1941)做出了重要贡献. ”—阿尔伯特·爱因斯坦著, 相对论的意义, 郝建纲、刘道军译, 上海科技教育出版社, 2001, p. 57.从数学建模的角度看, 广义相对论讨论的中心问题是引力理论, 其基础是以下两个假设: 1. (等效原理)惯性力场与引力场的动力学效应是局部不可分辨的，(或说引力和非惯性系中的惯性力等效)；2. (广义相对性原理) 一切参考系都是平权的，换言之，客观的真实的物理规律应该在任意坐标变换下形式不变——广义协变性(即一切物理定律在所有参考系[无论是惯性的或非惯性的]中都具有相同的形式)。
ReliabilityEngineeringandSystemSafety91(2006)992–1007
Reliability Engineering and System Safety 91(2006)992–1007Multi-objective optimization using genetic algorithms:A tutorialAbdullah Konak a,Ã,David W.Coit b ,Alice E.Smith caInformation Sciences and Technology,Penn State Berks,USA bDepartment of Industrial and Systems Engineering,Rutgers University cDepartment of Industrial and Systems Engineering,Auburn UniversityAvailable online 9January 2006AbstractMulti-objective formulations are realistic models for many complex engineering optimization problems.In many real-life problems,objectives under consideration conflict with each other,and optimizing a particular solution with respect to a single objective can result in unacceptable results with respect to the other objectives.A reasonable solution to a multi-objective problem is to investigate a set of solutions,each of which satisfies the objectives at an acceptable level without being dominated by any other solution.In this paper,an overview and tutorial is presented describing genetic algorithms (GA)developed specifically for problems with multiple objectives.They differ primarily from traditional GA by using specialized fitness functions and introducing methods to promote solution diversity.r 2005Elsevier Ltd.All rights reserved.1.IntroductionThe objective of this paper is present an overview and tutorial of multiple-objective optimization methods using genetic algorithms (GA).For multiple-objective problems,the objectives are generally conflicting,preventing simulta-neous optimization of each objective.Many,or even most,real engineering problems actually do have multiple-objectives,i.e.,minimize cost,maximize performance,maximize reliability,etc.These are difficult but realistic problems.GA are a popular meta-heuristic that is particularly well-suited for this class of problems.Tradi-tional GA are customized to accommodate multi-objective problems by using specialized fitness functions and introducing methods to promote solution diversity.There are two general approaches to multiple-objective optimization.One is to combine the individual objective functions into a single composite function or move all but one objective to the constraint set.In the former case,determination of a single objective is possible with methods such as utility theory,weighted sum method,etc.,but theproblem lies in the proper selection of the weights or utility functions to characterize the decision-maker’s preferences.In practice,it can be very difficult to precisely and accurately select these weights,even for someone familiar with the problem pounding this drawback is that scaling amongst objectives is needed and small perturbations in the weights can sometimes lead to quite different solutions.In the latter case,the problem is that to move objectives to the constraint set,a constraining value must be established for each of these former objectives.This can be rather arbitrary.In both cases,an optimization method would return a single solution rather than a set of solutions that can be examined for trade-offs.For this reason,decision-makers often prefer a set of good solutions considering the multiple objectives.The second general approach is to determine an entire Pareto optimal solution set or a representative subset.A Pareto optimal set is a set of solutions that are nondominated with respect to each other.While moving from one Pareto solution to another,there is always a certain amount of sacrifice in one objective(s)to achieve a certain amount of gain in the other(s).Pareto optimal solution sets are often preferred to single solutions because they can be practical when considering real-life problems/locate/ress0951-8320/$-see front matter r 2005Elsevier Ltd.All rights reserved.doi:10.1016/j.ress.2005.11.018ÃCorresponding author.E-mail address:konak@ (A.Konak).since thefinal solution of the decision-maker is always a trade-off.Pareto optimal sets can be of varied sizes,but the size of the Pareto set usually increases with the increase in the number of objectives.2.Multi-objective optimization formulationConsider a decision-maker who wishes to optimize K objectives such that the objectives are non-commensurable and the decision-maker has no clear preference of the objectives relative to each other.Without loss of generality, all objectives are of the minimization type—a minimization type objective can be converted to a maximization type by multiplying negative one.A minimization multi-objective decision problem with K objectives is defined as follows: Given an n-dimensional decision variable vector x¼{x1,y,x n}in the solution space X,find a vector x* that minimizes a given set of K objective functions z(x*)¼{z1(x*),y,z K(x*)}.The solution space X is gen-erally restricted by a series of constraints,such as g j(x*)¼b j for j¼1,y,m,and bounds on the decision variables.In many real-life problems,objectives under considera-tion conflict with each other.Hence,optimizing x with respect to a single objective often results in unacceptable results with respect to the other objectives.Therefore,a perfect multi-objective solution that simultaneously opti-mizes each objective function is almost impossible.A reasonable solution to a multi-objective problem is to investigate a set of solutions,each of which satisfies the objectives at an acceptable level without being dominated by any other solution.If all objective functions are for minimization,a feasible solution x is said to dominate another feasible solution y (x1y),if and only if,z i(x)p z i(y)for i¼1,y,K and z j(x)o z j(y)for least one objective function j.A solution is said to be Pareto optimal if it is not dominated by any other solution in the solution space.A Pareto optimal solution cannot be improved with respect to any objective without worsening at least one other objective.The set of all feasible non-dominated solutions in X is referred to as the Pareto optimal set,and for a given Pareto optimal set,the corresponding objective function values in the objective space are called the Pareto front.For many problems,the number of Pareto optimal solutions is enormous(perhaps infinite).The ultimate goal of a multi-objective optimization algorithm is to identify solutions in the Pareto optimal set.However,identifying the entire Pareto optimal set, for many multi-objective problems,is practically impos-sible due to its size.In addition,for many problems, especially for combinatorial optimization problems,proof of solution optimality is computationally infeasible.There-fore,a practical approach to multi-objective optimization is to investigate a set of solutions(the best-known Pareto set)that represent the Pareto optimal set as well as possible.With these concerns in mind,a multi-objective optimization approach should achieve the following three conflicting goals[1]:1.The best-known Pareto front should be as close aspossible to the true Pareto front.Ideally,the best-known Pareto set should be a subset of the Pareto optimal set.2.Solutions in the best-known Pareto set should beuniformly distributed and diverse over of the Pareto front in order to provide the decision-maker a true picture of trade-offs.3.The best-known Pareto front should capture the wholespectrum of the Pareto front.This requires investigating solutions at the extreme ends of the objective function space.For a given computational time limit,thefirst goal is best served by focusing(intensifying)the search on a particular region of the Pareto front.On the contrary,the second goal demands the search effort to be uniformly distributed over the Pareto front.The third goal aims at extending the Pareto front at both ends,exploring new extreme solutions.This paper presents common approaches used in multi-objective GA to attain these three conflicting goals while solving a multi-objective optimization problem.3.Genetic algorithmsThe concept of GA was developed by Holland and his colleagues in the1960s and1970s[2].GA are inspired by the evolutionist theory explaining the origin of species.In nature,weak and unfit species within their environment are faced with extinction by natural selection.The strong ones have greater opportunity to pass their genes to future generations via reproduction.In the long run,species carrying the correct combination in their genes become dominant in their population.Sometimes,during the slow process of evolution,random changes may occur in genes. If these changes provide additional advantages in the challenge for survival,new species evolve from the old ones.Unsuccessful changes are eliminated by natural selection.In GA terminology,a solution vector x A X is called an individual or a chromosome.Chromosomes are made of discrete units called genes.Each gene controls one or more features of the chromosome.In the original implementa-tion of GA by Holland,genes are assumed to be binary digits.In later implementations,more varied gene types have been introduced.Normally,a chromosome corre-sponds to a unique solution x in the solution space.This requires a mapping mechanism between the solution space and the chromosomes.This mapping is called an encoding. In fact,GA work on the encoding of a problem,not on the problem itself.GA operate with a collection of chromosomes,called a population.The population is normally randomly initia-lized.As the search evolves,the population includesfitterA.Konak et al./Reliability Engineering and System Safety91(2006)992–1007993andfitter solutions,and eventually it converges,meaning that it is dominated by a single solution.Holland also presented a proof of convergence(the schema theorem)to the global optimum where chromosomes are binary vectors.GA use two operators to generate new solutions from existing ones:crossover and mutation.The crossover operator is the most important operator of GA.In crossover,generally two chromosomes,called parents,are combined together to form new chromosomes,called offspring.The parents are selected among existing chromo-somes in the population with preference towardsfitness so that offspring is expected to inherit good genes which make the parentsfitter.By iteratively applying the crossover operator,genes of good chromosomes are expected to appear more frequently in the population,eventually leading to convergence to an overall good solution.The mutation operator introduces random changes into characteristics of chromosomes.Mutation is generally applied at the gene level.In typical GA implementations, the mutation rate(probability of changing the properties of a gene)is very small and depends on the length of the chromosome.Therefore,the new chromosome produced by mutation will not be very different from the original one.Mutation plays a critical role in GA.As discussed earlier,crossover leads the population to converge by making the chromosomes in the population alike.Muta-tion reintroduces genetic diversity back into the population and assists the search escape from local optima. Reproduction involves selection of chromosomes for the next generation.In the most general case,thefitness of an individual determines the probability of its survival for the next generation.There are different selection procedures in GA depending on how thefitness values are used. Proportional selection,ranking,and tournament selection are the most popular selection procedures.The procedure of a generic GA[3]is given as follows:Step1:Set t¼1.Randomly generate N solutions to form thefirst population,P1.Evaluate thefitness of solutions in P1.Step2:Crossover:Generate an offspring population Q t as follows:2.1.Choose two solutions x and y from P t based onthefitness values.ing a crossover operator,generate offspringand add them to Q t.Step3:Mutation:Mutate each solution x A Q t with a predefined mutation rate.Step4:Fitness assignment:Evaluate and assign afitness value to each solution x A Q t based on its objective function value and infeasibility.Step5:Selection:Select N solutions from Q t based on theirfitness and copy them to P t+1.Step6:If the stopping criterion is satisfied,terminate the search and return to the current population,else,set t¼t+1go to Step2.4.Multi-objective GABeing a population-based approach,GA are well suited to solve multi-objective optimization problems.A generic single-objective GA can be modified tofind a set of multiple non-dominated solutions in a single run.The ability of GA to simultaneously search different regions of a solution space makes it possible tofind a diverse set of solutions for difficult problems with non-convex,discon-tinuous,and multi-modal solutions spaces.The crossover operator of GA may exploit structures of good solutions with respect to different objectives to create new non-dominated solutions in unexplored parts of the Pareto front.In addition,most multi-objective GA do not require the user to prioritize,scale,or weigh objectives.Therefore, GA have been the most popular heuristic approach to multi-objective design and optimization problems.Jones et al.[4]reported that90%of the approaches to multi-objective optimization aimed to approximate the true Pareto front for the underlying problem.A majority of these used a meta-heuristic technique,and70%of all meta-heuristics approaches were based on evolutionary ap-proaches.Thefirst multi-objective GA,called vector evaluated GA (or VEGA),was proposed by Schaffer[5].Afterwards, several multi-objective evolutionary algorithms were devel-oped including Multi-objective Genetic Algorithm (MOGA)[6],Niched Pareto Genetic Algorithm(NPGA) [7],Weight-based Genetic Algorithm(WBGA)[8],Ran-dom Weighted Genetic Algorithm(RWGA)[9],Nondomi-nated Sorting Genetic Algorithm(NSGA)[10],Strength Pareto Evolutionary Algorithm(SPEA)[11],improved SPEA(SPEA2)[12],Pareto-Archived Evolution Strategy (PAES)[13],Pareto Envelope-based Selection Algorithm (PESA)[14],Region-based Selection in Evolutionary Multiobjective Optimization(PESA-II)[15],Fast Non-dominated Sorting Genetic Algorithm(NSGA-II)[16], Multi-objective Evolutionary Algorithm(MEA)[17], Micro-GA[18],Rank-Density Based Genetic Algorithm (RDGA)[19],and Dynamic Multi-objective Evolutionary Algorithm(DMOEA)[20].Note that although there are many variations of multi-objective GA in the literature, these cited GA are well-known and credible algorithms that have been used in many applications and their performances were tested in several comparative studies. Several survey papers[1,11,21–27]have been published on evolutionary multi-objective optimization.Coello lists more than2000references in his website[28].Generally, multi-objective GA differ based on theirfitness assign-ment procedure,elitisim,or diversification approaches.In Table1,highlights of the well-known multi-objective with their advantages and disadvantages are given.Most survey papers on multi-objective evolutionary approaches intro-duce and compare different algorithms.This paper takes a different course and focuses on important issues while designing a multi-objective GA and describes common techniques used in multi-objective GA to attain the threeA.Konak et al./Reliability Engineering and System Safety91(2006)992–1007 994goals in multi-objective optimization.This approach is also taken in the survey paper by Zitzler et al.[1].However,the discussion in this paper is aimed at introducing the components of multi-objective GA to researchers and practitioners without a background on the multi-objective GA.It is also import to note that although several of the state-of-the-art algorithms exist as cited above,many researchers that applied multi-objective GA to their problems have preferred to design their own customized algorithms by adapting strategies from various multi-objective GA.This observation is another motivation for introducing the components of multi-objective GA rather than focusing on several algorithms.However,the pseudo-code for some of the well-known multi-objective GA are also provided in order to demonstrate how these proce-dures are incorporated within a multi-objective GA.Table1A list of well-known multi-objective GAAlgorithm Fitness assignment Diversity mechanism Elitism ExternalpopulationAdvantages DisadvantagesVEGA[5]Each subpopulation isevaluated with respectto a differentobjective No No No First MOGAStraightforwardimplementationTend converge to theextreme of each objectiveMOGA[6]Pareto ranking Fitness sharing byniching No No Simple extension of singleobjective GAUsually slowconvergenceProblems related to nichesize parameterWBGA[8]Weighted average ofnormalized objectives Niching No No Simple extension of singleobjective GADifficulties in nonconvexobjective function space Predefined weightsNPGA[7]Nofitnessassignment,tournament selection Niche count as tie-breaker in tournamentselectionNo No Very simple selectionprocess with tournamentselectionProblems related to nichesize parameterExtra parameter fortournament selectionRWGA[9]Weighted average ofnormalized objectives Randomly assignedweightsYes Yes Efficient and easyimplementDifficulties in nonconvexobjective function spacePESA[14]Nofitness assignment Cell-based density Pure elitist Yes Easy to implement Performance depends oncell sizesComputationally efficientPrior information neededabout objective spacePAES[29]Pareto dominance isused to replace aparent if offspringdominates Cell-based density astie breaker betweenoffspring and parentYes Yes Random mutation hill-climbing strategyNot a population basedapproachEasy to implement Performance depends oncell sizesComputationally efficientNSGA[10]Ranking based onnon-dominationsorting Fitness sharing bynichingNo No Fast convergence Problems related to nichesize parameterNSGA-II[30]Ranking based onnon-dominationsorting Crowding distance Yes No Single parameter(N)Crowding distance worksin objective space onlyWell testedEfficientSPEA[11]Raking based on theexternal archive ofnon-dominatedsolutions Clustering to truncateexternal populationYes Yes Well tested Complex clusteringalgorithmNo parameter forclusteringSPEA-2[12]Strength ofdominators Density based on thek-th nearest neighborYes Yes Improved SPEA Computationallyexpensivefitness anddensity calculationMake sure extreme pointsare preservedRDGA[19]The problem reducedto bi-objectiveproblem with solutionrank and density asobjectives Forbidden region cell-based densityYes Yes Dynamic cell update More difficult toimplement than othersRobust with respect to thenumber of objectivesDMOEA[20]Cell-based ranking Adaptive cell-baseddensity Yes(implicitly)No Includes efficienttechniques to update celldensitiesMore difficult toimplement than othersAdaptive approaches toset GA parametersA.Konak et al./Reliability Engineering and System Safety91(2006)992–10079955.Design issues and components of multi-objective GA 5.1.Fitness functions5.1.1.Weighted sum approachesThe classical approach to solve a multi-objective optimization problem is to assign a weight w i to each normalized objective function z 0i ðx Þso that the problem is converted to a single objective problem with a scalar objective function as follows:min z ¼w 1z 01ðx Þþw 2z 02ðx ÞþÁÁÁþw k z 0k ðx Þ,(1)where z 0i ðx Þis the normalized objective function z i (x )and P w i ¼1.This approach is called the priori approach since the user is expected to provide the weights.Solving a problem with the objective function (1)for a given weight vector w ¼{w 1,w 2,y ,w k }yields a single solution,and if multiple solutions are desired,the problem must be solved multiple times with different weight combinations.The main difficulty with this approach is selecting a weight vector for each run.To automate this process;Hajela and Lin [8]proposed the WBGA for multi-objective optimization (WBGA-MO)in the WBGA-MO,each solution x i in the population uses a different weight vector w i ¼{w 1,w 2,y ,w k }in the calculation of the summed objective function (1).The weight vector w i is embedded within the chromosome of solution x i .Therefore,multiple solutions can be simulta-neously searched in a single run.In addition,weight vectors can be adjusted to promote diversity of the population.Other researchers [9,31]have proposed a MOGA based on a weighted sum of multiple objective functions where a normalized weight vector w i is randomly generated for each solution x i during the selection phase at each generation.This approach aims to stipulate multiple search directions in a single run without using additional parameters.The general procedure of the RWGA using random weights is given as follows [31]:Procedure RWGA:E ¼external archive to store non-dominated solutions found during the search so far;n E ¼number of elitist solutions immigrating from E to P in each generation.Step 1:Generate a random population.Step 2:Assign a fitness value to each solution x A P t by performing the following steps:Step 2.1:Generate a random number u k in [0,1]for each objective k ,k ¼1,y ,K.Step 2.2:Calculate the random weight of each objective k as w k ¼ð1=u k ÞP K i ¼1u i .Step 2.3:Calculate the fitness of the solution as f ðx Þ¼P K k ¼1w k z k ðx Þ.Step 3:Calculate the selection probability of each solutionx A P t as follows:p ðx Þ¼ðf ðx ÞÀf min ÞÀ1P y 2P t ðf ðy ÞÀf minÞwhere f min ¼min f f ðx Þj x 2P t g .Step 4:Select parents using the selection probabilities calculated in Step 3.Apply crossover on the selected parent pairs to create N offspring.Mutate offspring with a predefined mutation rate.Copy all offspring to P t +1.Update E if necessary.Step 5:Randomly remove n E solutions from P t +1and add the same number of solutions from E to P t +1.Step 6:If the stopping condition is not satisfied,set t ¼t þ1and go to Step 2.Otherwise,return to E .The main advantage of the weighted sum approach is a straightforward implementation.Since a single objective is used in fitness assignment,a single objective GA can be used with minimum modifications.In addition,this approach is computationally efficient.The main disadvan-tage of this approach is that not all Pareto-optimal solutions can be investigated when the true Pareto front is non-convex.Therefore,multi-objective GA based on the weighed sum approach have difficulty in finding solutions uniformly distributed over a non-convex trade-off surface [1].5.1.2.Altering objective functionsAs mentioned earlier,VEGA [5]is the first GA used to approximate the Pareto-optimal set by a set of non-dominated solutions.In VEGA,population P t is randomly divided into K equal sized sub-populations;P 1,P 2,y ,P K .Then,each solution in subpopulation P i is assigned a fitness value based on objective function z i .Solutions are selected from these subpopulations using proportional selection for crossover and mutation.Crossover and mutation are performed on the new population in the same way as for a single objective GA.Procedure VEGA:N S ¼subpopulation size (N S ¼N =K )Step 1:Start with a random initial population P 0.Set t ¼0.Step 2:If the stopping criterion is satisfied,return P t .Step 3:Randomly sort population P t .Step 4:For each objective k ,k ¼1,y K ,perform the following steps:Step 4.1:For i ¼1þðk 21ÞN S ;...;kN S ,assign fit-ness value f ðx i Þ¼z k ðx i Þto the i th solution in the sorted population.Step 4.2:Based on the fitness values assigned in Step 4.1,select N S solutions between the (1+(k À1)N S )th and (kN S )th solutions of the sorted population to create subpopulation P k .Step 5:Combine all subpopulations P 1,y ,P k and apply crossover and mutation on the combined population to create P t +1of size N .Set t ¼t þ1,go to Step 2.A similar approach to VEGA is to use only a single objective function which is randomly determined each time in the selection phase [32].The main advantage of the alternating objectives approach is easy to implement andA.Konak et al./Reliability Engineering and System Safety 91(2006)992–1007996computationally as efficient as a single-objective GA.In fact,this approach is a straightforward extension of a single objective GA to solve multi-objective problems.The major drawback of objective switching is that the popula-tion tends to converge to solutions which are superior in one objective,but poor at others.5.1.3.Pareto-ranking approachesPareto-ranking approaches explicitly utilize the concept of Pareto dominance in evaluatingfitness or assigning selection probability to solutions.The population is ranked according to a dominance rule,and then each solution is assigned afitness value based on its rank in the population, not its actual objective function value.Note that herein all objectives are assumed to be minimized.Therefore,a lower rank corresponds to a better solution in the following discussions.Thefirst Pareto ranking technique was proposed by Goldberg[3]as follows:Step1:Set i¼1and TP¼P.Step2:Identify non-dominated solutions in TP and assigned them set to F i.Step3:Set TP¼TPF i.If TP¼+go to Step4,else set i¼iþ1and go to Step2.Step4:For every solution x A P at generation t,assign rank r1ðx;tÞ¼i if x A F i.In the procedure above,F1,F2,y are called non-dominated fronts,and F1is the Pareto front of population P.NSGA[10]also classifies the population into non-dominated fronts using an algorithm similar to that given above.Then a dummyfitness value is assigned to each front using afitness sharing function such that the worst fitness value assigned to F i is better than the bestfitness value assigned to F i+1.NSGA-II[16],a more efficient algorithm,named the fast non-dominated-sort algorithm, was developed to form non-dominated fronts.Fonseca and Fleming[6]used a slightly different rank assignment approach than the ranking based on non-dominated-fronts as follows:r2ðx;tÞ¼1þnqðx;tÞ;(2) where nq(x,t)is the number of solutions dominating solution x at generation t.This ranking method penalizes solutions located in the regions of the objective function space which are dominated(covered)by densely populated sections of the Pareto front.For example,in Fig.1b solution i is dominated by solutions c,d and e.Therefore,it is assigned a rank of4although it is in the same front with solutions f,g and h which are dominated by only a single solution.SPEA[11]uses a ranking procedure to assign better fitness values to non-dominated solutions at underrepre-sented regions of the objective space.In SPEA,an external list E of afixed size stores non-dominated solutions that have been investigated thus far during the search.For each solution y A E,a strength value is defined assðy;tÞ¼npðy;tÞN Pþ1,where npðy;tÞis the number solutions that y dominates in P.The rank r(y,t)of a solution y A E is assigned as r3ðy;tÞ¼sðy;tÞand the rank of a solution x A P is calculated asr3ðx;tÞ¼1þXy2E;y1xsðy;tÞ.Fig.1c illustrates an example of the SPEA ranking method.In the former two methods,all non-dominated solutions are assigned a rank of1.This method,however, favors solution a(in thefigure)over the other non-dominated solutions since it covers the least number of solutions in the objective function space.Therefore,a wide, uniformly distributed set of non-dominated solutions is encouraged.Accumulated ranking density strategy[19]also aims to penalize redundancy in the population due to overrepre-sentation.This ranking method is given asr4ðx;tÞ¼1þXy2P;y1xrðy;tÞ.To calculate the rank of a solution x,the rank of the solutions dominating this solution must be calculatedfirst. Fig.1d shows an example of this ranking method(based on r2).Using ranking method r4,solutions i,l and n are ranked higher than their counterparts at the same non-dominated front since the portion of the trade-off surface covering them is crowded by three nearby solutions c,d and e. Although some of the ranking approaches described in this section can be used directly to assignfitness values to individual solutions,they are usually combined with variousfitness sharing techniques to achieve the second goal in multi-objective optimization,finding a diverse and uniform Pareto front.5.2.Diversity:fitness assignment,fitness sharing,and nichingMaintaining a diverse population is an important consideration in multi-objective GA to obtain solutions uniformly distributed over the Pareto front.Without taking preventive measures,the population tends to form relatively few clusters in multi-objective GA.This phenom-enon is called genetic drift,and several approaches have been devised to prevent genetic drift as follows.5.2.1.Fitness sharingFitness sharing encourages the search in unexplored sections of a Pareto front by artificially reducingfitness of solutions in densely populated areas.To achieve this goal, densely populated areas are identified and a penaltyA.Konak et al./Reliability Engineering and System Safety91(2006)992–1007997。
橡胶参数
超弹性(2008-07-17 15:32:31)标签：ansys教育分类：ANSYS学习铁板材料特性：Ex=2e5 (泊松比)＝0.3 橡胶体材料特性：(泊松比)＝0.499fini/cler=180l1=185l2=74h1=6h2=50h3=182r1=10d=50/prep7et,1,56,,,1et,2,42,,,1et,3,48,,1et,4,58et,5,45et,6,49,,1keyopt,6,7,1keyopt,3,7,1r,1,10000,,0.5rect,0,l1,0,h1rect,0,l2,0,h2cyl4,0,h3,r,-90,,0 aovlap,allasel,s,loc,y,0,h1asel,r,loc,x,0,l2 aadd,allallsasel,s,loc,y,h1,h3 aadd,allallslsel,s,loc,x,0lsel,r,loc,y,0,h1lcom,allallslsel,s,loc,x,0lsel,r,loc,y,h1,h3 lcom,allallslsel,s,loc,y,h1lsel,r,loc,x,0,l2lcom,allallslsel,s,loc,y,h1,h2lsel,r,loc,x,l2*get,line1,line,,num,max allslsel,s,loc,x,l2,l1lsel,r,loc,y,h2,h3*get,line2,line,,num,max allslfillt,line1,line2,r1 allsal,1,4,3asel,s,loc,y,h1,h3 aadd,allallslsel,s,loc,x,l2,l1lsel,r,loc,y,h1+0.1,h3-0.1 lcom,allallslsel,s,loc,x,l2,l1lsel,r,loc,y,h1+0.1,h3-0.1 *get,line3,line,,num,max allskl,line3,0.12lsel,s,loc,x,0lsel,r,loc,y,h1,h3*get,line4,line,,num,max kl,line4,0.4kl,line4,0.7allslstr,5,7lstr,13,8asel,s,loc,y,h1,h3lsel,s,,,3,4asbl,all,allallsasel,s,loc,y,h1,h3aatt,1,1,1asel,s,loc,y,0,h1aatt,2,1,2allslesize,3,,,10lesize,8,,,12,2lesize,7,,,6lesize,12,,,1lesize,19,,,8mshkey,1amesh,5amesh,3amesh,2amesh,1amesh,6allslsel,s,,,19nsll,s,1cm,targ,nodeallslsel,s,,,6nsll,s,1cm,cont1,nodelsel,s,,,8nsll,s,1nsel,r,loc,y,h1,130 cm,cont2,nodeallscmsel,s,cont1 cmsel,a,cont2!cmsel,a,cont3cm,cont,nodeallstype,3mat,1real,1gcgen,cont,targallssave,hypelastic,db resume,hypelastic,db mp,nuxy,1,0.499 mp,ex,2,2e5mp,nuxy,2,0.3m=1.95m1=2.03n=1.05*dim,strn,,10,3*dim,strss,,10,3*dim,const,,5*dim,calc,,30*dim,sortss,,30*dim,sortsn,,30*dim,ffx,table,30,2*dim,ecalc,table,100*dim,xval,table,100strn(1,2)=-0.45,-0.4,-0.35,-0.3,-0.25,-0.2,-0.15,-0.1,-0.05strss(1,2)=-256,-128,-64,-32,-16,-8,-4,-2,-1*do,i,1,9strss(i,2)=strss(i,2) *mstrn(i,2)=strn(i,2)*m1*enddostrn(1,1)=0.0,0.2,0.4,0.6,0.8,1.0,1.2,1.6,2,3strss(1,1)=0.0,1,1.5,2.0,2.9,3.6,5,7.5,9.7,17strn(1,3)=0.0,0.05,0.2,0.3,0.4,.5,.6,.8,1,1.5strss(1,3)=0.0,0.7,1.5,2.0,2.9,3.6,5,7.5,9.7,17*do,i,1,10strss(i,3)=strss(i,3)* n*enddotb,mooney,,,,1*mooney,strn(1,1),strss(1,1),,const(1),calc(1),sortsn(1),sortss(1) *vfun,ffx(1,1),copy,sortss(1)*vfun,ffx(1,2),copy,calc(1)*vplot,strn(1),ffx(1),2*eval,1,2,const(1),-0.2,0,xval(1),ecalc(1)*vplot,xval(1),ecalc(1)fini/soluallsnsel,s,loc,y,0d,all,all,0allsnsel,s,loc,x,0d,all,ux,0d,all,uz,0allsnsel,s,loc,y,h3d,all,ux,0d,all,uz,0d,all,uy,-dallsantype,staticnlgeom,onnropt,,,onoutpr,all,alloutres,all,allautots,ontime,1deltim,0.03,0.01,0.3cnvtol,f,,0.02,2lnsrch,onpred,onallssolvefiniansys-Beam3二维弹性单元特性翻译工程应力与真实应力(2008-07-31 13:47:36)标签：ansys教育分类：ANSYS学习fini/cle/PREP7ET,1,plane182KEYOPT,1,3,1R,1,0.001, , , , , ,MP,EX,1,2.1E11 ! STEELMP,NUXY,1,0.3TB,BKIN,1,1 ! DEFINE NON-LINEAR MATERIAL PROPERTY FOR STEEL TBTEMP,0TBDATA,1,210e6,8.6e9BLC4,0, ,0.03,0.03NUMCMP,ALLAESIZE,1,0.003,allsamesh,allnsel,s,loc,y,0nsel,a,loc,y,0.05d,all,uynsel,s,loc,y,0.03nsel,a,loc,y,0.08D,all, ,0.00009,, , ,Uy ! if >0.00003 material is yield;alls/SOLUNLGEOM, ON!According small strain theory 0.005 cause 0.3%= (0.00009/0.03) strain; but if we trun NLGEOM ON, the strain is 0.2996%=ln(1+0.00009/0.03)NSUBST, 40, 100, 40OUTRES, ALL, 1SOLVE/POST1SET, LASTPLNSOL, EPTO,Y, 0,1.0 ! the max total strain value is 0.2996%/repl/POST26RFORCE, 2, 22, f, y, FY_2PLVAR, 2ANSOL,4,22,EPEL,y,EPELy_2ANSOL,5,22,EPPL,y,EPPLy_4ANSOL,6,22,S,y,Sy_4ADD,7,4,5, , , , ,1,1,1,/AXLAB,X, DEFLECTION/AXLAB,Y, Stress/GRID,1XVAR,7PLVAR,6超级大变形(2008-07-15 15:56:19)标签：ansys分类：ANSYS学习fini/cle/PREP7lsize=600hsize=24l=135e-3h=6e-3p=-10e-3!定义单元!ET,1,VISCO108ET,1,SHELL181ET,2,MASS21KEYOPT,2,1,0KEYOPT,2,2,0KEYOPT,2,3,2KEYOPT,1,1,0KEYOPT,1,2,0KEYOPT,1,3,0KEYOPT,1,5,0KEYOPT,1,6,0KEYOPT,1,7,0KEYOPT,1,8,0KEYOPT,1,9,0KEYOPT,1,11,0!实常数R,1,0.3e-3, , , , , , !板厚RMORE, , , ,RMORERMORE, ,R,2,(5.3e-3)/(hsize-1), !集中质量!定义材料MPTEMP,,,,,,,,MPTEMP,1,0MPDATA,EX,1,,2.06e11MPDATA,PRXY,1,,0.3 MPTEMP,,,,,,,,MPTEMP,1,0MPDATA,DENS,1,,7900!板尺寸BLC4,p, ,l,h!单元大小LESIZE,1, , ,lsize, , , , ,1LESIZE,2, , ,hsize, , , , ,1 MSHAPE,0,2DMSHKEY,1AMESH,1TYPE,2REAL,2nsel,s,loc,x,l+pnsel,r,extnsel,u,loc,y,0nsel,u,loc,y,h*get,n_min,node,,num,minn_num=n_min*do,i,1,hsize-1E,n_numn_num=ndnext(n_num)*enddo!约束DL,4,,ALLEPLOTallssave!求解/SOLUANTYPE,4 !瞬态大变形TRNOPT,FULLLUMPM,0NLGEOM,1NSUBST,20,100,10 !子步数OUTRES,ERASEOUTRES,esol,LASTOUTRES,nsol,LASTAUTOTS,1lnsrch,1PRED,ONSSTIF,1KBC,0!cnvtol,f,0.05,,, !收敛容差!cnvtol,u,0.05,,,!冲击波加载J=1*do,i,1.1e-5,1.1e-3,1.1e-4 !载荷步数可在此改time,iacel,,,9810*sin(2854.5*i) !冲击波形lswrite,jj=j+1*enddo*do,i,1.1e-3+1e-3,30e-3,1e-3 !载荷步数可在此改time,iNSUBST,50,100,10acel,,,-361.94*sin(108.7*(i-1.1e-3)) !冲击波形lswrite,jj=j+1*enddoacel,,,0*do,i,30e-3+1e-3,0.05,1e-3 !载荷步数可在此改time,iNSUBST,50,100,10lswrite,jj=j+1*enddoj=j-1save!求解lssolve,1,J,1,Ansys疲劳算例(2008-02-22 13:38:45)标签：ansys 教育! ***************环境设置***************/units,si/title, Fatigue analysis of cylinder with flat head! ***************参数设定***************Di=1000 ! 筒体内径t=20 ! 筒体厚度hc=nint(4*sqrt(Di/2*t)/10)*10 ! 模型中筒体长度tp=60 ! 平板封头厚度r1=10 ! 平板封头外测过渡圆弧半径r2=10 ! 平板封头内侧应力释放槽圆弧半径exx=2e5 ! 材料弹性模量mu=0.3 ! 材料泊松比p1=2 ! 最高工作压力p3=2.88 ! 水压试验压力n1=2e4 ! 最高/最低压力循环次数n2=5 ! 水压试验次数! ***************前处理***************/prep7et,1,82 ! 设定单元类型keyopt,1,3,1 ! 设定周对称选项mp,ex,1,exx ! 定义材料弹性模量mp,nuxy,1,mu ! 定义材料泊松比! ******* 建立模型*******k,1,0,0 ! 定义关键点k,2,Di/2+t,,k,3,Di/2+t,-(tp+hc)k,4,Di/2,-(tp+hc)k,5,Di/2,-tpk,6,Di/2-r2,-tp ! 定义应力释放槽圆弧中心关键点k,7,0,-tpl,1,2 ! 生成线l,2,3l,3,4l,4,5l,5,7l,7,1LFILLT,1,2,r1 ! 生成外测过渡圆弧al,all ! 生成子午面CYL4, kx(6),ky(6), r2,180 ! 生成应力释放槽面域ASBA,1,2 ! 面相减wprot,,,90 ! 旋转工作平面wpoff,,,kx(6)-3*r2 ! 移动工作平面asbw,all ! 用工作平面切割子午面wprot,,90 ! 旋转工作平面wpoff,,,tp+r2 ! 移动工作平面asbw,all ! 用工作平面切割子午面esize,5 ! 设定单元尺寸MSHKEY,1 ! 设定映射剖分amesh,1 ! 映射剖分面1amesh,3 ! 映射剖分面3esize,2 ! 设定单元尺寸MSHKEY,0 ! 设定自由剖分amesh,4 ! 自由剖分面4fini ! 退出前处理! ***************求解***************/solu ! 筒体端部施加轴向约束dl,3,,uy ! 筒体端部施加轴向约束dl,6,,symm ! 平板封头对称面施加对称约束time,1 ! 载荷步1lsel,s,,,8 ! 选择内表面各线段lsel,a,,,11,13lsel,a,,,15cm,lcom1,line ! 生成内表面线组件SFL,all,PRES,p1, ! 内表面施加内压alls ! 全选solve ! 求解fini ! 退出求解器! ***************后处理***************/post1 ! 进入后处理FTSIZE,1,2,2, ! 设定疲劳评定的位置数、事件数及载荷数FP,1,1e1,2e1,5e1,1e2,2e2,5e2 ! 根据疲劳曲线输入S－N数据FP,7,1e3,2e3,5e3,1e4,2e4,5e4FP,13,1e5,2e5,5e5,1e6, ,FP,19, ,FP,21,4000,2828,1897,1414,1069,724FP,27,572,441,331,262,214,159FP,33,138,114,93.1,86.2, ,FP,39, ,! ****** 水压试验循环******fs,4760,1,1,1,0,0,0,0,0,0 ! 储存节点4760对应其第一载荷的应力set,1,last ! 读入第一载荷步数据FSNODE,4760,1,2 ! 储存节点4760对应其第二载荷的应力fe,1,n2,p3/p1 ! 设定事件循环次数及载荷比例系数! ****** 最高/最低压力循环******fs,4760,2,1,1,0,0,0,0,0,0 ! 储存节点4760对应其第一载荷的应力set,1,last ! 读入第一载荷步数据FSNODE,4760,2,2 ! 储存节点4760对应其第二载荷的应力FE,2,n1,1, ! 设定事件循环次数及载荷比例系数FTCALC,1 ! 进行疲劳计算（并记录使用系数）fini!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~计算结果如下：PERFORM FATIGUE CALCULATION AT LOCATION 1 NODE 0*** POST1 FATIGUE CALCULATION ***LOCA TION 1 NODE 4760事件1：****** 水压试验循环******EVENT/LOADS 1 1 AND 1 2PRODUCE ALTERNA TING SI (SALT) = 285.16（应力幅值）CYCLES USED/ALLOWED = 5.000/7779（实际循环数/许用循环数）= PARTIAL USAGE （局部损伤)=0.00064事件2：****** 最高/最低压力循环******EVENT/LOADS 2 1 AND 2 2PRODUCE ALTERNA TING SI (SALT) = 198.03 WITH TEMP = 0.0000CYCLES USED/ALLOWED = 0.2000E+05/ 0.2541E+05 = PARTIAL USAGE = 0.78719CUMULATIVE FATIGUE USAGE = 0.78784注意：285/198=P3/P1，应力与载荷成线性关系节点4760的S1,S3分别为：395，-1.2；应力幅值=(S1-S3)/2=(395-(-1.2))/2198对应的许用循环数0.2541E＋05是通过S－N(214->20000 ,159->50000)曲线插值出来的.下面的命令流进行的是一个简单的二维焊接分析, 利用ANSYS单元生死和热-结构耦合分析功能进!行焊接过程仿真, 计算焊接过程中的温度分布和应力分布以及冷却后的焊缝残余应力。