| Frame structures are widely used in various engineering fields,such as civil engineering,construction,machinery,and aerospace.For quantitative analysis of such structures,the finite element method(FEM)is the most popular approach.However,the accuracy of FEM is absolutely controlled by the mesh,so that the remeshing is required when the accuracy needs to be adjusted,resulting in a low efficiency.In addition,creating a satisfactory finite element mesh with local refinement for solving problems with local steep gradients is usually a time-consuming task and even cannot be carried out without manual work.It is particularly true in the analysis of problems such as the crack identification,in which the mesh is needed to be adjusted dynamically.Thus,the combination of FEM and the wavelet analysis has been developed and successfully applied to analyzing frame structures.In such a method,the global and local accuracy can be readily adjusted by changing the global or local resolution level of elements.Moreover,since the wavelet holds the low-pass filtering property,the displacement approximation used in the wavelet finite element method(WFEM)can accurately represent low-frequency signals in a numerical sense.As a result,the WFEM obviously surpassed the traditional FEM in vibration and buckling analysis.However,the existing wavelet approximations are lack of C~1continuity when approximating functions piecewise,because the wavelet analysis is originally defined on the whole interval.This will result in a terrible difficulty in imposing the essential boundary condition and continuity condition between adjacent elements.Thus,some additional techniques such as the Lagrange multiplier method and transformation matrix are required in the existing WFEMs,which will cause the increase of calculation and potential stability problem.In order to address this tough issue,a wavelet multiresolution interpolation with C~1continuity is first proposed in this work.Based on such a wavelet interpolation,a multiresolution beam element is then developed to study frame structures.And a systematic investigation on the accuracy and convergence of the proposed method is also conducted by theoretical analysis and numerical test.Firstly,a boundary extension technique based on Hermite polynomial interpolation is proposed.It effectively suppresses numerical instabilities near boundaries caused by series truncation in wavelet analysis.Subsequently,combining this new technology with the classic 4th-order interpolation wavelet,a wavelet multi-resolution segmented interpolation format with global C~1continuity is established,and its tight error estimation is given.Unlike the wavelet approximation used in existing methods,the improved wavelet multiresolution C~1interpolation does not need to calculate any inverse matrix in analysis.This ensures the efficiency and stability of the algorithm.At the same time,it completely retains the vanishing moment and low-pass filtering characteristics of wavelet analysis.In the numerical sense,it has the ability to accurately reconstruct low-order polynomials and low-frequency signals.It also has multi-resolution analysis capabilities that increase the level of local resolution.In addition,this dissertation also gives the error estimation of wavelet multiresolution C~0interpolation based on Lagrangian boundary extension technique and interpolation wavelet.The precise calculation methods of the interpolation wavelet derivatives,integrals and connection coefficients are given.Based on wavelet multiresolution C~0interpolation,the wavelet multiresolution element of tension and compression bars is constructed for truss structures.Then,the algorithm to analyze the deformation and free vibration of the truss structure is given.The truss structure composed of constant-section bar and variable-section bar under various loads and boundary conditions is quantitatively studied.The numerical results show that the WFEM can effectively study the statics and dynamics of truss structures.Especially for modal analysis,the given 4th-order wavelet element can obtain high-precision results for higher-order modes with fewer degrees of freedom.Furthermore,the numerical results also show that the wavelet method can capture local large gradients very efficiently and stably because of the multiresolution analysis.Subsequently,based on the developed wavelet multiresolution C~1interpolation,the solution format of Euler Bernoulli beams is established.Similarly,the wavelet multiresolution bar element is constructed.In this element,the configurations of the axial and bending degrees of freedom are completely independent.The optimal configuration of degrees of freedom under different working conditions is realized.At the same time,the angle degrees of freedom are imposed only at the two endpoints for applying the essential boundary condition and continuity condition,and the deflection degrees of freedom are set inside the element.Therefore,this effectively reduces the degrees of freedom of the element compared to the traditional FEM.The results show that the wavelet multiresolution bar element established in this paper can quantitatively analyze the deformation,free vibration and eigenvalue buckling problems of rigid frame structures very effectively.Especially for modal analysis and buckling analysis,the WFEM proposed in this paper can obtain higher precision results with fewer degrees of freedom than traditional FEM.This is because the adopted wavelet multiresolution interpolation can efficiently reconstruct low-frequency signals.For example,for the free vibration of a simply supported beam acting with an axial force,when the former only uses 37 degrees of freedom,the relative errors of the first three natural frequencies can reach 7×10-5,2×10-4and 6×10-4.When the latter uses 52 degrees of freedom,the relative errors are only 3×10-3,5×10-3and 1×10-2,respectively.In addition,the wavelet bar element established in this paper can adjust the overall and local analysis accuracy by simply adjusting the element resolution level and multi-resolution mode. |
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