طراحي پي با استفاده از بهينه‌سازي توپولوژيك

Foundation Design Using Topology Optimization

چكيده- در اين مقاله، طراحي بهينه پي با استفاده هم‌زمان از بهينه‌سازي توپولوژيك و بهينه‌سازي ابعادي بررسي شده است. مدل استفاده‌شده، از المان‌هاي صفحه‌اي براي پي گسترده، المان‌هاي حجمي براي خاك و المان‌هاي فشاري براي اتصال خاك و پي تشكيل شده است. تحليل و طراحي به صورت الاستيك و سه‌بعدي تحت يك بارگذاري معين انجام مي‌شود. با توجه به وابستگي تعداد متغيرهاي طراحي به تعداد المان‌هاي محدود در فضاي طراحي (كه معمولاً زياد است)، استفاده از روش‌هاي كلاسيك در بهينه‌سازي توپولوژيك سازه‌ها عملي‌ست. از اين رو در اين مقاله از روشي نوين براي ايجاد توپولوژي بهينه استفاده مي‌شود كه ايده اصلي آن بر پايه‌ي كارايي المان‌هاي واسط بين خاك و پي در انتقال نيروهاي فشاري به خاك است. اين الگوريتم جديد، المان‌هايي از پي كه تحت فشار اندك يا صفر قرار دارند را حذف مي‌كند. بهينه‌سازي كلي از دو فاز مجزاي بهينه‌سازي توپولوژيك و بهينه‌سازي ابعادي تشكيل شده است. بهينه‌سازي ابعادي شامل تعيين ضخامت و درصد آرماتورهاي خمشي بهينه پي است. روند بهينه‌سازي ابعادي و توپولوژي در پي هم تكرار ميشود تا ديگر پيشرفتي در بهينه‌سازي بوجود نيايد. كارايي روش پيشنهادشده در اين مقاله با حل چهار نمونه عددي بررسي شده است.

In this paper a methodology for optimal design of foundation of structures has been proposed. The structural model consists of foundation, soil and foundation-soil connectors. The 3D finite element model comprises three different finite elements, i.e. homogeneous and isotropic plate elements with three degrees of freedom for each node for mat foundation, brick elements with three degrees of freedom for each node for soil and some compression-only members for connection of foundation to soil named Gap elements which connect the foundation to the soil in three ways. To facilitate the optimization algorithm, a 3D analysis is performed in the elastic range under one design loading. Due to numerous finite elements and accordingly considerable number of design variables, the classical topology optimization algorithms become inefficient for solution of the problem. Therefore in this paper an innovative topology optimization algorithm, which does work based on efficiency of foundation elements, is proposed. This new algorithm cuts out the foundation elements that do not bear (or bear low value) compression under design load(s). The overall optimization methodology consists of two phase optimization: namely topology optimization and size optimization which are carried out in 12 steps. Objective function is determined as minimizing the cost of the foundation consisiting of concrete and reinforcement costs. Design variables are existence or nonexistence of the foundation elements, thickness of the foundation and the percentage of the reinforcement. Some of the foundation elements are indicated as constant elements during the elimination process. Each of the other foundation elements should be removed if the gap element connecting the element to soil is recognized as nonbearing element under the design loading. Constraints of the optimization problem are bending moment and shear force in foundation and compression stress of the soil. After every topology optimization, an analysis is done and a size optimization algorithm is used to proportion the thickness of the foundation and corresponding reinforcement. The consequent topology and size optimization processes are repeated until no enhancement is obtained in the design. The capabilities of the proposed method have been shown by four design examples. Results show that, depending on the constraints of the optimization problem, the optimal topology of the initial mat foundation varies from common topologies such as strip foundation to unusual topology of foundation. Based on the results, the optimal topology of the foundation depends on the design loading(s) and constraints of the optimization problem.