Yapısal dinamik analizlerin sonlu elemanlar cevaplarının süper elemanlar ve alt yapılara bölme ile iyileştirilmesi

Improvement of structural dynamic finite elements responses by using super elements and structuring

Özet

Bu çalışmada yapısal dinamik problemlerin analizlerinde sonlu elemanlar yöntemi ile elde edilen cevaplarının süper elemanlar ile iyileştirilmesi incelenmiştir. Geliştirilen yeni model derecesi dü-şürme (MDD) yaklaşımları ile olumlu sonuçlar elde edilmiştir. Bu yeni yaklaşımlar sistem zaman cevaplarının eşitliğini sağlayacak serbestlik derecelerinin aktif serbestlik dereceleri olarak MDD uygulanmış sistemin içine taşınması temeline dayanmaktadır. Hazırlanan Matlab kodları ile gelişti-rilen model derecesi düşürme yöntemi örnek yapılara uygulanmış ve alınan zaman cevabı sonuçla-rı, orijinal modelin sonuçları ile karşılaştırılmıştır. Tüm bu karşılaştırmalarda Rayleigh sönüm mo-deline uygun farklı katılık orantılı sönüm matrisi durumları dikkate alınmış ve farklı model derece-leri için hesaplama zamanları açısından sonuçlar değerlendirilmiştir. Ayrıca farklı MDD teknikleri ile elde edilen sistem toplam enerji miktarı orijinal modelin toplam enerjisi ile karşılaştırılmıştır. Çalışmada alt yapılara bölme (AYB)-“Substructuring” yöntemi de incelenmiştir. Geliştirilen Mat-lab kodu ile plak sistemleri üzerinde AYB uygulamları yapılmış, sonuçlar orijinal model cevabı ile karşılaştırılmıştır. Oluşturulan süper elemanların kullanılması ile de hesaplama zamanları önemli ölçüde düşürülmüştür. Ancak süper eleman oluşturmaya harcanan zamanın yüksek olduğu ve AYB yönteminin geliştirilen MDD yönteminden daha kötü performans gösterdiği görülmüştür. Yapılan tüm karşılaştırmalar sonucunda çalışmada incelenen yöntemlerin çok düşük olan sönüm durumları dışında, orijinal model ile uyumlu zaman ve frekans cevapları verdiği, MDD uygulanmış sistemin orijinal model ile aynı enerji seviyelerinde bulunduğu ve karşılaştırılan diğer MDD yöntemlerinden çok daha az hesaplama zamanına ihtiyaç duyulduğu görülmüştür.

Abstract

In this study, the responses obtained by the Finite Element Analyses of structural dynamic problems are studied. With the developed model order reduc-tion approach, some improvements have been gained. The developed model order reduction meth-od is essentially based on the equality of the total energies of both original and reduced systems and therefore it is named as “Equality of the total ener-gies”. In the new approach, respecting the degrees of freedoms (DOF)s that give responses conforming with original system is assumed as the main criteria for selection of active DOFs. In this study, the main significant step is the selection of the active DOFs according to their influence to the system time re-sponse. The groups of active are application points of the external loadings and the DOFs having high stress or deflection values maximum because they are very important to reveal the right system re-sponse. For better accuracy, DOFs close to loading application points are also transferred into reduced system. Heuristic method had been also used for the selection of the active DOFs by some trials and er-rors to get better consistency between the original system response and reduced system response. De-veloped model order reduction method was applied to a sample 2D truss system and cantilever plate by some Matlab codes, and the results of the reduction analyses are compared with the original system re-sponses in response to various loadings, eg. İm-pulse, step and sinusoidal inputs. In the same Matlab codes, time responses of some other model order reduction methods –Forward differences, Newmark integration and impulse response invari-ant (IRI)- are also implemented to the same original model. Various Rayleigh damping models for the sample system are taken into account in all these comparisons and the results of analyses for different model orders are evaluated in terms of calculation time and accuracy. Frequency domain responses of the original and the reduced systems by different approaches are also compared with each other based on Bode diagrams by considering the stiffness proportional damping. In parallel, the total energy levels of the reduced systems are also compared with the original system energy level. While auditing of the system responses and Bode diagrams, it is noted that the compatibility of both the original and the reduced systems increases in parallel with the stiffness proportional damping value. In evaluation of the results, it could be expressed that the change in stiffness proportional damping value does not af-fect the calculation time. While comparing the total energies of original and reduced systems, it is ob-served that their consistency is good enough to state that both systems have the same system characteris-tic. Bode diagrams which reflect frequency response of a system, are assumed to be the main criteria to evaluate the performance of the reduced system for both high and low frequencies. It is concluded in general that when the number of an element type used in the analyses decreases, the amount of the DOFs which could be omitted from the original sys-tem increases and also the order of the reduced sys-tem decreases by keeping the same expected accura-cy. As a result of observation, the limit of the model order reduction depends on the selected elemen type. Another conclusion is that general behavior of the damped or undamped systems do not affect the selected element numbers for the reduced system. In the study, substructing method is also taken into considerations. By using Matlab codes, substructur-ing is implemented into a plate system and the re-sults are compared with the original system. It is observed that the calculation time is extremely re-duced by using super elements Nevertheless, since the time for calculation of super elements is long, it is seen that substructuring exhibits worse perfor-mance than developed model order reduction meth-od. After all comparisons, it is noted that the developed method gives compatible time and frequency re-sponses with the original model except too low stiff-ness proportional dampings. It is also observed that the energy level of the reduced system obtained by this new model order reduction method is almost the same with the energy level of the original model. Finally, the most advantageous feature of equality of the total energies method was determined on the calculation time and it needs less calculation time than the other compared model order reduction methods during all examples carried out in the study.