內(nèi)域介質(zhì)波動數(shù)值模擬的若干研究
發(fā)布時間：2018-11-28 20:42
【摘要】：內(nèi)域介質(zhì)波動數(shù)值模擬是大型結(jié)構(gòu)-基礎(chǔ)相互作用分析中的一個重要方面。本文基于有限元-有限差分法及ABAQUS/Explicit積分格式,研究離散網(wǎng)格中波動的傳播規(guī)律,提出計算精度的分析方法及改善數(shù)值頻散的方案。主要工作有:1.對于ABAQUS/Explicit積分格式假定的正確性進(jìn)行驗證;建立了有阻尼體系顯式積分格式計算精度的一種新方法,并應(yīng)用于ABAQUS/Explicit積分格式的精度分析。推得單自由度體系質(zhì)點振動的周期延長率、峰值遞減率隨頻率、阻尼比及計算時間步長變化的解析式。取工程常用的阻尼比,對積分格式的計算精度進(jìn)行了定量分析。2.基于ABAQUS/Explicit積分格式應(yīng)用于顯式有限元-有限差分法時的遞推形式,推導(dǎo)出傳遞函數(shù)矩陣,給出離散有限元網(wǎng)格中波動的傳播規(guī)律。無阻尼情況下存在截止頻率,大于截止頻率的波動成分無法傳播;有阻尼情況時,小于Nyquist頻率的波動成分可以在離散網(wǎng)格中傳播,接近以及大于截止頻率ωu的波動成分的幅值會迅速衰減。時間積分步長At取值越小或者空間步距Ax取值越大,截止頻率ωu越小,波動中的高頻成分衰減越快,積分格式的能耗越大。3.通過與理論解進(jìn)行對比來評價算法的計算精度,結(jié)果表明:低頻區(qū)時數(shù)值計算結(jié)果和理論解很接近;阻尼系數(shù)β取值比較小時,數(shù)值解的幅值衰減項與理論解擬合程度不高,隨著β取值增大,計算結(jié)果精確性提高。4.通過計算相速度和群速度分析了一維有阻尼體系數(shù)值模擬的頻散效應(yīng)。采用集中質(zhì)量矩陣時,在滿足穩(wěn)定性的條件下,時間步長取值越小,對波動中高頻成分的抑制越明顯,但波動中的頻散也會增大;采用一致質(zhì)量矩陣時,時間步長At取值越小,數(shù)值模擬的結(jié)果精度越高,波動的頻散越小。采用不同的質(zhì)量矩陣形式時時間步長的選擇很重要。一致質(zhì)量矩陣時,阻尼系數(shù)β越大,數(shù)值模擬的波動頻散越大,這與理論上波動頻散規(guī)律相同,但是β對集中質(zhì)量矩陣的頻散影響相反。5.將集中質(zhì)量矩陣與一致質(zhì)量矩陣進(jìn)行線性組合,這種方法能提高計算結(jié)果的精度并且有效地壓制頻散,最優(yōu)的線性組合系數(shù)為0.5。
[Abstract]:Numerical simulation of internal medium wave is an important aspect in the analysis of large-scale structure-foundation interaction. Based on the finite element finite difference method and ABAQUS/Explicit integral scheme, the propagation law of wave in discrete mesh is studied, and the analytical method of calculation accuracy and the scheme of improving numerical dispersion are put forward in this paper. The main work is: 1. The correctness of the assumption of ABAQUS/Explicit integral scheme is verified and a new method for calculating the accuracy of explicit integral scheme of damped system is established and applied to the accuracy analysis of ABAQUS/Explicit integral scheme. An analytical formula for the variation of the period prolongation rate of particle vibration and the peak decline rate with frequency damping ratio and calculating time step of single degree of freedom system is derived. Taking the damping ratio commonly used in engineering, the calculation accuracy of integral scheme is quantitatively analyzed. 2. Based on the recursive form of ABAQUS/Explicit integral scheme applied to explicit finite-difference finite element method, the transfer function matrix is derived, and the propagation law of wave in discrete finite element mesh is given. In the case of undamped, the fluctuation component larger than the cutoff frequency can not propagate because of the existence of cutoff frequency. In the case of damping, the wave components smaller than the Nyquist frequency can propagate in the discrete grid, and the amplitude of the wave components close to and larger than the cutoff frequency 蠅 u will decay rapidly. The smaller the At value of the time integral step or the greater the Ax value of the spatial step, the smaller the cutoff frequency 蠅 u, the faster the attenuation of the high frequency component in the fluctuation, and the greater the energy consumption of the integral scheme. The accuracy of the algorithm is evaluated by comparing with the theoretical solution. The results show that the numerical results in the low frequency region are very close to the theoretical solution. When the damping coefficient 尾 is small, the amplitude attenuation term of the numerical solution is not well fitted with the theoretical solution. With the increase of 尾 value, the accuracy of the calculation results is improved. 4. The dispersion effect of numerical simulation of one-dimensional damped system is analyzed by calculating phase velocity and group velocity. When the lumped mass matrix is adopted, the smaller the time step is, the more obvious the suppression of the high frequency component in the wave is, but the frequency dispersion in the fluctuation will increase. When the uniform mass matrix is adopted, the smaller the time step At is, the higher the accuracy of the numerical simulation results is and the smaller the dispersion of the fluctuation is. The choice of time step is very important when adopting different mass matrix forms. In the case of uniform mass matrix, the larger the damping coefficient 尾, the larger the wave dispersion of numerical simulation, which is the same as the theoretical wave dispersion law, but the effect of 尾 on the dispersion of lumped mass matrix is opposite. 5. The linear combination of lumped mass matrix and uniform mass matrix can improve the accuracy of the calculation results and suppress dispersion effectively. The optimal linear combination coefficient is 0.5.
【學(xué)位授予單位】：中國水利水電科學(xué)研究院
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O302
[Abstract]:Numerical simulation of internal medium wave is an important aspect in the analysis of large-scale structure-foundation interaction. Based on the finite element finite difference method and ABAQUS/Explicit integral scheme, the propagation law of wave in discrete mesh is studied, and the analytical method of calculation accuracy and the scheme of improving numerical dispersion are put forward in this paper. The main work is: 1. The correctness of the assumption of ABAQUS/Explicit integral scheme is verified and a new method for calculating the accuracy of explicit integral scheme of damped system is established and applied to the accuracy analysis of ABAQUS/Explicit integral scheme. An analytical formula for the variation of the period prolongation rate of particle vibration and the peak decline rate with frequency damping ratio and calculating time step of single degree of freedom system is derived. Taking the damping ratio commonly used in engineering, the calculation accuracy of integral scheme is quantitatively analyzed. 2. Based on the recursive form of ABAQUS/Explicit integral scheme applied to explicit finite-difference finite element method, the transfer function matrix is derived, and the propagation law of wave in discrete finite element mesh is given. In the case of undamped, the fluctuation component larger than the cutoff frequency can not propagate because of the existence of cutoff frequency. In the case of damping, the wave components smaller than the Nyquist frequency can propagate in the discrete grid, and the amplitude of the wave components close to and larger than the cutoff frequency 蠅 u will decay rapidly. The smaller the At value of the time integral step or the greater the Ax value of the spatial step, the smaller the cutoff frequency 蠅 u, the faster the attenuation of the high frequency component in the fluctuation, and the greater the energy consumption of the integral scheme. The accuracy of the algorithm is evaluated by comparing with the theoretical solution. The results show that the numerical results in the low frequency region are very close to the theoretical solution. When the damping coefficient 尾 is small, the amplitude attenuation term of the numerical solution is not well fitted with the theoretical solution. With the increase of 尾 value, the accuracy of the calculation results is improved. 4. The dispersion effect of numerical simulation of one-dimensional damped system is analyzed by calculating phase velocity and group velocity. When the lumped mass matrix is adopted, the smaller the time step is, the more obvious the suppression of the high frequency component in the wave is, but the frequency dispersion in the fluctuation will increase. When the uniform mass matrix is adopted, the smaller the time step At is, the higher the accuracy of the numerical simulation results is and the smaller the dispersion of the fluctuation is. The choice of time step is very important when adopting different mass matrix forms. In the case of uniform mass matrix, the larger the damping coefficient 尾, the larger the wave dispersion of numerical simulation, which is the same as the theoretical wave dispersion law, but the effect of 尾 on the dispersion of lumped mass matrix is opposite. 5. The linear combination of lumped mass matrix and uniform mass matrix can improve the accuracy of the calculation results and suppress dispersion effectively. The optimal linear combination coefficient is 0.5.
【學(xué)位授予單位】：中國水利水電科學(xué)研究院
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O302
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