A Numerical Study on Hydraulic Fracturing Problems via the Proper Generalized Decomposition Method

The hydraulic fracturing is a nonlinear,fluid-solid coupling and transient problem,in most cases it is always time-consuming to simulate this process numerically.In recent years,although many numerical methods were proposed to settle this problem,most of them still require a large amount of computer resources.Thus it is a high demand to develop more efficient numerical approaches to achieve the real-time monitoring of the fracture geometry during the hydraulic fracturing treatment.In this study,a reduced order modeling technique namely Proper Generalized Decomposition(PGD),is applied to accelerate the simulations of the transient,non-linear coupled system of hydraulic fracturing problem,to match this extremely tight response time constraint.The separability of the solution in space and time dimensions is studied for a simplified model problem.The solid and fluid equations are coupled explicitly by inverting the solid discrete problem,and a simple iterative procedure to handle the non-linear characteristic of the hydraulic fracturing problem is proposed in this work.Numeral validation illustrates that the results of PGD match well with these of standard finite element method in terms o f fracture opening and fluid pressure in the hydro-fracture.Moreover,after the off-line calculations,the numerical results can be obtained in real time.

Numerical Tools for the Control of the Unsteady Heating of an Airfoil

This paper concerns the real time control of the boundary layer on an aircraft wing. This new approach consists in heating the surface in an unsteady regime using electrically resistant strips embedded in the wing skin. The control of the boundary layer＇s separation and transition point will provide a reduction in friction drag, and hence a reduction in fuel consumption. This new method consists in applying the required thermal power in the different strips in order to ensure the desired temperatures on the aircraft wing. We also have to determine the optimum size of these strips （length, width and distance between two strips）. This implies finding the best mathematical model corresponding to the physics enabling us to facilitate the calculation for any type of material used for the wings. Secondly, the heating being unsteady, and, as during a flight the flow conditions or the ambient temperatures vary, the thermal power needed changes and must be chosen as fast as possible in order to ensure optimal operating conditions.

Direct design to stress mapping for cellular structures

This paper aims to instantly predict within any accuracy the stress distribution of cellular structures under parametric design,including the shapes or distributions of the cell geometries,or the magnitudes of external loadings.A classical model reduction technique has to balance the simulation accuracy and interaction speed,and has difficulty achieving this goal.We achieve this by computing offline a design-to-stress mapping that ultimately expresses the stress distribution as an explicit function in terms of its design parameters.The mapping is determined as a solution to an extended finite element analysis problem in a high-dimension space,including both the spatial coordinates and the design parameters.The well-known curse of dimensionality intrinsic to the high-dimension problem is(partly)resolved through a spatial separation using two main techniques.First,the target mapping takes a reduced form as a sum of the products of separated one-variable functions,extending the proper generalized decomposition technique.Second,the simulation problem in a varied computation domain is reformulated as that in a fixed-domain,taking an integration function as the sum of the products of separated one-variable functions,in combination with high-order singular value decomposition.Extensive 2D and 3D examples are shown to demonstrate the approach’s performance.