بهبود شبكه نقاط كنترل در تحليل ايزوژيومتريك سازه هاي متقارن محوري با استفاده از روش هاي تخمين خطا مبتني بر بازيافت تنش

Control point grid improvement in isogeometric analysis of

روش ابداعي اين پژوهش، با استفاده از برآورد خطاي مبتني بر بازيافت تنش، به حل تطبيقي و بهبود شبكه مسايل متقارن محوري در - تحليل ايزوژيومتريك مي پردازد. بدين منظور، بعد از محاسبه نرم خطاي انرژي، مقادير تخميني خطا در مجاورت نقاط كنترلي به عنوان گراديان حرارتي به المان هاي رابط بين اين نقاط اختصاص مي يابد. پس از تحليل شبكه المان هاي فرضي كه دچار تغيير دما شد هاند آرايش جديدي از نقاط كنترلي و بردارهاي گرهي حاصل ميشود؛ كه انتظار مي رود به كارگيري اين روند در چند سيكل در تحليل ايزوژيومتريك مسايل متقارن محوري منجر به توزيع بهتر خطا در دامنه و در نتيجه حصول شبكه اي بهينه براي محاسبه انتگرال ها شود. براي بررسي كارايي و نشان دادن افزايش دقت در نتايج تحليل مسايل متقارن محوري به مدل سازي عددي دو مسيله الاستيسيته با شرايط تقارن محوري، كه داراي حل تحليلي هستند، پرداخته شده است. نتايج به دست آمده نشان مي دهد كه روش بهبود شبكه ارايه شده در كاهش ميزان خطا موثر بوده و مي تواند براي افزايش دقت نتايج تحليل ايزوژيومتريك مسايل متقارن محوري مورد استفاده قرار گيرد.

This research is devoted to the adaptive solution of axisymmetric problems in isogeometric analysis. Actually several numerical methods have been devised for solving the governing partial differential equations that are encountered in real life engineering problems, e.g. finite difference, finite elements, boundary elements and the family of mesh-free methods. One of the most recent of these methods is the isogeometric analysis method, which still is in its early stages of development. Isogeometric analysis has a few interesting features that presents itself as a potential substitute to the other numerical techniques. For this purpose, splines and some new versions of them, i.e. non-uniform rational B-splines (NURBS) and T-splines, are usually employed. Accepting the point that by using proper versions of splines, one is able to define complex geometries, any component of a field variable which satisfies a governing partial differential equation might be imagined as a surface. After using isogeometric analysis concept estimating the errors is necessary for improvement of solution. Since all numerical methods are approximate, the obtained solutions are inevitably erroneous and hence the question of the degree of reliability of the obtained results is a main concern of scientists and engineers. Today the procedures available for error estimation are essentially reduced to two kinds, the residual error estimators and the use of self-equilibrating patches which has recently attracted more attention. In the second approach a recovery process is followed to obtain more accurate representation of the unknowns. In this research we are employed this recovery procedure. Since the purpose of error estimation is to use it for improvement of solution, after carrying out analysis and estimation of errors, we are interested in increasing accuracy of solution by reducing and leveling the distribution of errors. This procedure is usually referred to as adaptive solution or adaptivity. Therefore, for an adaptive solution, first the location and magnitude of errors are evaluated by using an error estimation tool and then somehow it is tried to have a uniform distribution of errors. This might be achieved by mesh refinement which is called, h adaptivity in the finite element literature. An approach which falls into category of h refinement, is r method. In this approach the number of discretization points remains constant but their location varies according to the solution error distribution. This is the method used in this research in isogeometric method. Therefore in this paper, after the calculation of the error norm, the estimated value of error in the vicinity of each control point is assigned to the neighboring members of a hypothetical structure as an artificial thermal gradient. By analysis of this network a new placement of control points is obtained. Repeating isogeometric analysis will eventually lead to a better distribution of errors in the domain of the problem. To demonstrate the performance and efficiency of the proposed method, two axisymmetric elasticity problems with available analytical solutions are considered. The obtained results indicate that this innovative approach is effective in reducing errors of axisymmetric problems and can be employed for improving the accuracy in the context of the isogeometric analysis method.