Java编程实现有限元方法

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"本书《Programming Finite Elements in Java》由Gennadiy Nikishkov撰写，主要探讨如何使用Java编程语言实现有限元计算方法。作者来自日本会津若松市的会津大学，这本书由Springer London出版，具有国际标准书号（ISBN）和电子书号（e-ISBN）。书中涵盖了与有限元相关的理论知识以及实际的Java编程实践，适用于对有限元方法感兴趣的工程师、研究人员和学生。" 在有限元方法中，我们首先需要理解其基本概念，它是数值分析领域的一种强大工具，用于解决各种工程和物理问题，特别是那些涉及偏微分方程的问题。有限元将连续区域划分为许多互不重叠的子区域（有限元），然后对每个子区域内的方程进行近似求解，最终通过集成这些局部解来获得整个区域的全局解。这种方法尤其适合处理复杂几何形状和非线性问题。在Java环境下实现有限元计算，我们需要关注以下几个关键知识点： 1. **数据结构**：为了存储和操作复杂的几何信息以及有限元网格，通常需要设计和实现高效的数据结构，如节点、元素和边界条件的表示。 2. **线性代数**：有限元方法涉及大量的矩阵运算，如求解线性系统。因此，熟悉Java中的线性代数库，如Apache Commons Math或JAMA，是必不可少的。 3. **数值积分**：在有限元方法中，需要进行积分以计算元素贡献和组装全局矩阵。这可能涉及到高斯积分或其他数值积分技术。 4. **非线性处理**：对于非线性问题，可能需要迭代求解过程，如牛顿-拉弗森方法，Java编程时需考虑迭代算法的实现。 5. **求解器**：选择合适的线性系统求解器，如直接法（如LU分解）或迭代法（如CG、GMRES等），并了解如何在Java中集成这些求解器。 6. **图形可视化**：使用Java 2D或Java 3D库可以实现结果的图形化展示，这对于理解和验证计算结果至关重要。 7. **错误和精度控制**：理解和实现误差估计以及收敛性检验，确保计算的准确性和可靠性。 8. **并行计算**：利用Java的多线程特性或Java并行计算框架，如Fork/Join框架，可优化大规模有限元问题的求解效率。 9. **输入/输出**：设计用户友好的输入接口读取问题的定义，以及输出结果文件的格式化，可能需要使用如XML、JSON等数据交换格式。 10. **软件工程实践**：遵循良好的编程规范，使用版本控制工具（如Git），编写测试用例，以及文档编写，都是实现一个健壮的有限元软件所必需的。这本书将深入讨论以上各个主题，并提供实际的Java代码示例，帮助读者将理论知识转化为实际的程序实现。通过阅读和实践，读者可以掌握如何在Java环境中构建有限元求解器，解决实际工程问题。

xvi Contents

B Data for Mesh Generation ......................................383

B.1 Mesh-generation Modules ...................................383

B.2 Rectangular Mesh Block . ...................................384

B.3 Mesh Inside Eight-node Macroelement ........................384

B.4 Three-dimensionalMeshbySweeping.........................384

B.5 ReadingMeshfromFile.....................................385

B.6 Writing Mesh to File . .......................................385

B.7 CopyingMesh.............................................385

B.8 MeshTransformations ......................................385

B.9 ConnectingTwoMeshBlocks................................386

C Data for Visualizer .............................................387

C.1 VisualizationData..........................................387

C.2 Input Data.................................................387

D Example of Problem Solution ....................................389

D.1 ProblemStatement .........................................389

D.2 MeshGeneration...........................................390

D.3 ProblemSolution...........................................391

D.4 Visualization ..............................................393

References .........................................................397

Index .............................................................399

4 1 Introduction

1. Discretize the domain. The ﬁrst step is to divide a solution region (domain) into

ﬁnite elements connected at nodes. Because of the large amount of data, the ﬁnite

element mesh is typically generated by a preprocessor program. The description

of a mesh consists of several arrays, the main of which are nodal coordinates and

element connectivities.

2. Determine interpolation functions. Interpolation functions are used to inter-

polate the ﬁeld variables over the element. Usually, polynomials are selected as

interpolation functions. The degree of the polynomial depends on the number of

nodes belonging to the element. Interpolation functions are commonly called shape

functions since they are also used for deﬁnition of the element shape.

3. Compute the element matrices and vectors. The matrix equation for the ﬁ-

nite element is established that relates the nodal values of the unknown function to

known parameters. For this task different approaches can be used; the most conve-

nient are: the variational approach and the Galerkin method.

4. Assemble the element equations. To ﬁnd the global equation system for the

whole solution region we must assemble all the element equations. In other words

we must properly combine local element equations for all elements used for dis-

cretization. Element connectivities are used for the assembly process. Before solu-

tion, boundary conditions (which are not accounted for in the element equations)

should be imposed.

5. Solve the global equation system. The ﬁnite element global equation system is

typically sparse, symmetric and positive-deﬁnite. Direct and iterative methods can

be used for solution. The nodal values of the sought function are produced as a result

of the solution.

6. Compute additional results. In many cases we need to calculate additional pa-

rameters. For example, in mechanical problems strains and stresses are of interest

in addition to displacements, which are obtained after solution of the global equa-

tion system. While the primary result function (displacements) is continuous, its

derivatives (strains and stresses) have discontinuities at element boundaries.

1.2 Formulation of Finite Element Equations

Several approaches can be used to transform the physical formulation of the prob-

lem to its ﬁnite element discrete analog. If the physical formulation of the problem

is known as a differential equation then the most popular method of its ﬁnite ele-

ment formulation is the Galerkin method. If the physical problem can be formulated

as minimization of a functional then variational formulation of the ﬁnite element

equations is usually used.

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请问怎么更改报错信息 Error in numbers2colors(datTraits, signed = FALSE) : 'lim' must be finite. In addition: Warning messages: 1: In FUN(newX[, i], ...) : no non-missing arguments to min; returning Inf 2: In FUN(newX[, i], ...) : no non-missing arguments to max; returning -Inf