区间B样条小波有限元在含裂纹结构及经典层合板中的应用
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摘要
随着工程结构日趋复杂化、材料多样化及计算机技术的发展，数值计算在工程结构设计和分析中占据着重要地位，已成为现代科学研究的主要手段之一。含裂纹结构在裂纹尖端处应力具有1/(?)r奇异性，为获得较高计算精度，传统有限元需在裂纹尖端处划分细致的网格，裂纹扩展时，原有的网格需要重新划分，使数值计算精度和效率大幅度降低，严重影响裂纹参数的准确识别。工程断裂数值分析中，由于环境变化、加工和装配误差等的影响，设计参数常存在不确定性，设计分析时如果忽略这些因素的影响，将会得到不切合实际的计算结果，模糊数学是解决不确定性问题的主要工具。复合材料层合板由于具有比强度比刚度高、耐腐蚀性能好及材料性能可设计性等优点，被广泛应用于卫星发射、国防建设等尖端科技领域，然而复合材料层合板材料各向异性、结构呈层性及边缘应力分布复杂性等给传统有限元数值分析带来了很大困难。因此，如何提高含裂纹结构及复合材料层合板数值计算精度和效率，准确识别裂纹损伤参数，以及如何利用模糊理论解决断裂数值分析中不确定性问题是当今数值计算领域研究的热点问题。
基于上述原因，本文开展了区间B样条小波（B-spline wavelet on the interval,BSWI）有限元的应用研究。BSWI有限元是近年来发展起来的一种新的数值计算方法，以区间B样条小波尺度函数或小波函数作为插值函数，具有待定系数少、逼近精度高、局部化性能强和多分辨分析等特性，可弥补传统有限元在处理裂纹奇异性问题和复合材料层合板数值计算时的不足。目前，BSWI有限元已广泛应用于偏微分方程数值求解、故障诊断、变载荷构件应力分析和几何非线性等方面，而在断裂参数数值计算、不确定性问题分析和复合材料层合板应力分析等方面研究还较少。鉴于此，本文通过对BSWI有限元的研究，将其应用到含裂纹结构、不确定性问题分析和经典层合板数值计算中，提出了一系列以提高含裂纹结构和经典层合板数值计算精度和效率的算法，具体有：
1、利用区间B样条小波良好的局部化特性，能以较少的单元和自由度数最大限度地逼近待求函数，本文将BSWI有限元引入到断裂参数数值计算中，提出了基于BSWI有限元的虚拟裂纹闭合法，建立了含裂纹结构BSWI有限元模型，利用BSWI哑节点断裂单元提取裂纹尖端附近节点信息，应用虚拟裂纹闭合法计算应力强度因子，数值算例结果表明此算法计算简单，具有较高的计算精度和效率，为实际工程中含裂纹结构复杂数值计算提供了新方法。
2、为解决工程实际中载荷和材料参数等不确定性给断裂数值计算带来的影响，将模糊理论、BSWI有限元理论和断裂力学相结合，提出了含裂纹结构的模糊BSWI有限元法，建立了含裂纹结构模糊BSWI有限元模型，推导出模糊BSWI有限元平衡方程，利用λ水平截集及区间数分解定理求解模糊BSWI有限元平衡方程，数值算例结果表明该方法更为真实、准确地反映了结构响应的变化情况，为工程实际中处理含不确定参数断裂数值计算提供了一种新途径。
3、微小裂纹的存在和扩展，往往会导致重大灾难性事故的发生，因而，预知微小裂纹的存在并定量识别其参数是工程实际中的重要课题，裂纹的出现，将引起结构局部刚度的改变，从而改变结构的动力学特性，本文从线弹性断裂力学角度出发，利用弹簧单元模拟裂纹引起的局部柔度变化，建立了裂纹梁BSWI有限元辨识模型，利用BSWI有限元裂纹故障诊断算法对矩形截面裂纹梁进行裂纹参数定量识别，数值模拟和实验研究验证了该算法的可行性和准确性。
4、为解决经典层合板拉-剪、弯-扭、拉剪-弯扭耦合效应给传统有限元数值计算带来的困难，本文将BSWI有限元与经典层合板理论相结合，提出了经典层合板应力分析的BSWI有限元法，首次采用同尺度不同阶数BSWI尺度函数张量积插值，构造出BSWI经典层合板单元，推导了经典层合板BSWI有限元平衡方程，数值算例表明了该算法的有效性和高效性。
With complexity of engineering structures increasing, diversity of materials andthe development of computer technology, numerical calculation plays a fundamentaland important role in the design and analysis of engineering structures, it has becomeone of the essential means of modern scientific research. Because of stress singularityat the crack tip, extremely refined meshes have to be used at the crack-tip and newmeshes need to be re-divided as the crack propagates, which causes lower numericalaccuracy and efficiency for the traditional finite element method, severely influencingthe accurate identification of crack parameters. During numerical analysis ofengineering, design parameters have uncertainties due to the environmental changesand errors caused by machining and assembling. If these uncertain factors are ignored,we may get unreasonable calculation results, fuzzy mathematics is the main tool tosolve the uncertainty problem. Composite laminates are widely used in the satellitelaunch, national defense construction and other advanced technology areas because ofthe advantages of high specific strength and high specific stiffness, good corrosionresistance and material properties. However, the anisotropy of the material, layer ofstructure as well as the complexity of the edge stress distribution makes it difficult forthe traditional finite element method. Therefore, the researches on how to improve thenumerical calculation accuracy and efficiency of structures with cracks and compositelaminates plates，and combine fuzzy mathematics with numerical analysis of fracturehave become one of the focused topics. As a new numerical calculation method,BSWI employs wavelet scaling function or wavelet function of BSWI as interpolationfunction, it has a lot of good properties, such as less undetermined coefficients, highapproximation accuracy, strong localized performance and multi-resolution analysis,all of which can make up the defects for traditional finite element to solve the crackproblems and the laminated composite plates. At present, BSWI finite element methodhas been widely used in lots of fields, such as the numerical solution of partialdifferential equations, fault diagnosis, stress analysis of variable load and geometricnonlinear problems, however, the studies of numerical calculation of the fractureparameter, uncertainties analysis and stress analysis of laminated composite plates are less. In view of this, BSWI finite element method is applied to numerical calculationof the structure with cracks and laminated composite plates, a series of novelapproaches are proposed here to improve the numerical calculation accuracy andefficiency.
BSWI functions have lots of good localized features, which can use fewer unitsand the number of degrees of freedom to maximize the approximation of the unknownfunction, as a result, virtual crack closure technique is proposed by applying BSWIfinite element to numerical calculation of the fracture parameter. Moreover, BSWIfinite element model with cracks is established, the node displacements at crack tip areattached by dummy node to fracture elements, stress intensity factor is calculated byvirtual crack closure technique, comparison the calculated results with those byANSYS shows that the method put forward in this paper is easy to calculate, withhigher accuracy and less elements, provides a new way for engineering fractureanalysis of structure with cracks.
In order to solve the impact caused by uncertainties of loads and materials inengineering practices for numerical calculation of fracture, fuzzy finite elementmethod based on B-spline wavelet on the interval for structure with cracks is proposedby combining fuzzy theory and BSWI finite element theory with fracture mechanics.The BSWI fuzzy finite element model with cracks is established, the BSWI fuzzyequilibrium equations is deduced, specially, a strategy, Level cutting collection λ andthe interval number decomposition method are adopted to solve the intervalequilibrium equations. The analysis results demonstrate that the proposed method canaccurately reflect changes in structural response with fewer elements, moreover, thisapproach provides a new way for engineering fracture analysis with uncertaintiescomplex structures.
The appearance and extension of tiny cracks in the structure may lead tocatastrophic accidents. Therefore, predicting the presence of tiny cracks andquantitative identification of parameters has important values to practical engineering.Local stiffness changes of the structure will be caused by emergence of crack,following by the modification of structural dynamics characteristics, based on this,from the view of linear elastic fracture mechanics, the identification model of crackbeam of BSWI finite element is established by using the spring element simulation ofcrack quantitative identification of cracked beam，which is finished by using crack fault diagnosis algorithm of BSWI finite element, the numerical simulation andexperimental study verify the accuracy and feasibility of the algorithm.
Laminated composite plates have coupling effects of tension-shear, bending-torsion, tension with shear-bending and torsion with the complexity of the edgestress distribution, which are difficult to calculate for the traditional finite element. Tosolve this problem, BSWI finite element methods is combined with the classicallaminated plate theory in this paper, the BSWI finite element method for stressanalysis of classical laminates is proposed. Using the same scale but the differentorder BSWI scaling function tensor product interpolation constructs the element ofBSWI classical laminated plate, so BSWI finite element equilibrium equation of theclassical laminated plate is derived, the numerical simulation verifies the feasibility ofthe algorithm and accuracy.
基于上述原因，本文开展了区间B样条小波（B-spline wavelet on the interval,BSWI）有限元的应用研究。BSWI有限元是近年来发展起来的一种新的数值计算方法，以区间B样条小波尺度函数或小波函数作为插值函数，具有待定系数少、逼近精度高、局部化性能强和多分辨分析等特性，可弥补传统有限元在处理裂纹奇异性问题和复合材料层合板数值计算时的不足。目前，BSWI有限元已广泛应用于偏微分方程数值求解、故障诊断、变载荷构件应力分析和几何非线性等方面，而在断裂参数数值计算、不确定性问题分析和复合材料层合板应力分析等方面研究还较少。鉴于此，本文通过对BSWI有限元的研究，将其应用到含裂纹结构、不确定性问题分析和经典层合板数值计算中，提出了一系列以提高含裂纹结构和经典层合板数值计算精度和效率的算法，具体有：
1、利用区间B样条小波良好的局部化特性，能以较少的单元和自由度数最大限度地逼近待求函数，本文将BSWI有限元引入到断裂参数数值计算中，提出了基于BSWI有限元的虚拟裂纹闭合法，建立了含裂纹结构BSWI有限元模型，利用BSWI哑节点断裂单元提取裂纹尖端附近节点信息，应用虚拟裂纹闭合法计算应力强度因子，数值算例结果表明此算法计算简单，具有较高的计算精度和效率，为实际工程中含裂纹结构复杂数值计算提供了新方法。
2、为解决工程实际中载荷和材料参数等不确定性给断裂数值计算带来的影响，将模糊理论、BSWI有限元理论和断裂力学相结合，提出了含裂纹结构的模糊BSWI有限元法，建立了含裂纹结构模糊BSWI有限元模型，推导出模糊BSWI有限元平衡方程，利用λ水平截集及区间数分解定理求解模糊BSWI有限元平衡方程，数值算例结果表明该方法更为真实、准确地反映了结构响应的变化情况，为工程实际中处理含不确定参数断裂数值计算提供了一种新途径。
3、微小裂纹的存在和扩展，往往会导致重大灾难性事故的发生，因而，预知微小裂纹的存在并定量识别其参数是工程实际中的重要课题，裂纹的出现，将引起结构局部刚度的改变，从而改变结构的动力学特性，本文从线弹性断裂力学角度出发，利用弹簧单元模拟裂纹引起的局部柔度变化，建立了裂纹梁BSWI有限元辨识模型，利用BSWI有限元裂纹故障诊断算法对矩形截面裂纹梁进行裂纹参数定量识别，数值模拟和实验研究验证了该算法的可行性和准确性。
4、为解决经典层合板拉-剪、弯-扭、拉剪-弯扭耦合效应给传统有限元数值计算带来的困难，本文将BSWI有限元与经典层合板理论相结合，提出了经典层合板应力分析的BSWI有限元法，首次采用同尺度不同阶数BSWI尺度函数张量积插值，构造出BSWI经典层合板单元，推导了经典层合板BSWI有限元平衡方程，数值算例表明了该算法的有效性和高效性。
With complexity of engineering structures increasing, diversity of materials andthe development of computer technology, numerical calculation plays a fundamentaland important role in the design and analysis of engineering structures, it has becomeone of the essential means of modern scientific research. Because of stress singularityat the crack tip, extremely refined meshes have to be used at the crack-tip and newmeshes need to be re-divided as the crack propagates, which causes lower numericalaccuracy and efficiency for the traditional finite element method, severely influencingthe accurate identification of crack parameters. During numerical analysis ofengineering, design parameters have uncertainties due to the environmental changesand errors caused by machining and assembling. If these uncertain factors are ignored,we may get unreasonable calculation results, fuzzy mathematics is the main tool tosolve the uncertainty problem. Composite laminates are widely used in the satellitelaunch, national defense construction and other advanced technology areas because ofthe advantages of high specific strength and high specific stiffness, good corrosionresistance and material properties. However, the anisotropy of the material, layer ofstructure as well as the complexity of the edge stress distribution makes it difficult forthe traditional finite element method. Therefore, the researches on how to improve thenumerical calculation accuracy and efficiency of structures with cracks and compositelaminates plates，and combine fuzzy mathematics with numerical analysis of fracturehave become one of the focused topics. As a new numerical calculation method,BSWI employs wavelet scaling function or wavelet function of BSWI as interpolationfunction, it has a lot of good properties, such as less undetermined coefficients, highapproximation accuracy, strong localized performance and multi-resolution analysis,all of which can make up the defects for traditional finite element to solve the crackproblems and the laminated composite plates. At present, BSWI finite element methodhas been widely used in lots of fields, such as the numerical solution of partialdifferential equations, fault diagnosis, stress analysis of variable load and geometricnonlinear problems, however, the studies of numerical calculation of the fractureparameter, uncertainties analysis and stress analysis of laminated composite plates are less. In view of this, BSWI finite element method is applied to numerical calculationof the structure with cracks and laminated composite plates, a series of novelapproaches are proposed here to improve the numerical calculation accuracy andefficiency.
BSWI functions have lots of good localized features, which can use fewer unitsand the number of degrees of freedom to maximize the approximation of the unknownfunction, as a result, virtual crack closure technique is proposed by applying BSWIfinite element to numerical calculation of the fracture parameter. Moreover, BSWIfinite element model with cracks is established, the node displacements at crack tip areattached by dummy node to fracture elements, stress intensity factor is calculated byvirtual crack closure technique, comparison the calculated results with those byANSYS shows that the method put forward in this paper is easy to calculate, withhigher accuracy and less elements, provides a new way for engineering fractureanalysis of structure with cracks.
In order to solve the impact caused by uncertainties of loads and materials inengineering practices for numerical calculation of fracture, fuzzy finite elementmethod based on B-spline wavelet on the interval for structure with cracks is proposedby combining fuzzy theory and BSWI finite element theory with fracture mechanics.The BSWI fuzzy finite element model with cracks is established, the BSWI fuzzyequilibrium equations is deduced, specially, a strategy, Level cutting collection λ andthe interval number decomposition method are adopted to solve the intervalequilibrium equations. The analysis results demonstrate that the proposed method canaccurately reflect changes in structural response with fewer elements, moreover, thisapproach provides a new way for engineering fracture analysis with uncertaintiescomplex structures.
The appearance and extension of tiny cracks in the structure may lead tocatastrophic accidents. Therefore, predicting the presence of tiny cracks andquantitative identification of parameters has important values to practical engineering.Local stiffness changes of the structure will be caused by emergence of crack,following by the modification of structural dynamics characteristics, based on this,from the view of linear elastic fracture mechanics, the identification model of crackbeam of BSWI finite element is established by using the spring element simulation ofcrack quantitative identification of cracked beam，which is finished by using crack fault diagnosis algorithm of BSWI finite element, the numerical simulation andexperimental study verify the accuracy and feasibility of the algorithm.
Laminated composite plates have coupling effects of tension-shear, bending-torsion, tension with shear-bending and torsion with the complexity of the edgestress distribution, which are difficult to calculate for the traditional finite element. Tosolve this problem, BSWI finite element methods is combined with the classicallaminated plate theory in this paper, the BSWI finite element method for stressanalysis of classical laminates is proposed. Using the same scale but the differentorder BSWI scaling function tensor product interpolation constructs the element ofBSWI classical laminated plate, so BSWI finite element equilibrium equation of theclassical laminated plate is derived, the numerical simulation verifies the feasibility ofthe algorithm and accuracy.
引文
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