A multi layer gridless area router is reported.Based on corner stitching,this router adopts tile expansion to explore path for each net.A heuristic method that penalizes nodes deviating from the destination is devise...A multi layer gridless area router is reported.Based on corner stitching,this router adopts tile expansion to explore path for each net.A heuristic method that penalizes nodes deviating from the destination is devised to accelerate the algorithm.Besides,an enhanced interval tree is used to manage the intermediate data structure.In order to improve the completion rate of routing,a new gridless rip up and rerouting algorithm is proposed.The experimental results indicate that the completion rate is improved after the rip up and reroute process and the speed of this algorithm is satisfactory.展开更多
Gridless method is developed for unsteady viscous flows involving moving boundaries. The point distri- bution of gridless method is implemented in an isotropic or anisotropic way according to the features of viscous f...Gridless method is developed for unsteady viscous flows involving moving boundaries. The point distri- bution of gridless method is implemented in an isotropic or anisotropic way according to the features of viscous flows. In the area far away from the body, the traditional cloud of isotropic points is used, while in the adjacent area, the cloud of anisotropic points is distributed. In this way, the point spacing normal to the wall can be small enough for simulating the boundary layer, and meanwhile, the total number of points in the computational do- main can be controlled due to large spacing in other tangential direction through the anisotropic way. A fast mov- ing technique of clouds of points at each time-step is presented based on the attenuation law of disturbed motion for unsteady flows involving moving boundaries. In the mentioned cloud of points, a uniform weighted least- square curve fit method is utilized to discretize the spatial derivatives of the Navier-Stokes equations. The pro- posed gridless method, coupled with a dual time-stepping method and the Spalart-Allmaras turbulence model, is implemented for the Navier-Stokes equations. The computational results of unsteady viscous flows around a NLR7301 airfoil with an oscillating flap and a pitching NACA0012 airfoil are presented in a good agreement with the available experimental data.展开更多
A fast hybrid algorithm based on gridless method coupled with finite volume method （FVM） is developed for the solution to Euler equations. Compared with pure gridless method, the efficiency of the hybrid algorithm i...A fast hybrid algorithm based on gridless method coupled with finite volume method （FVM） is developed for the solution to Euler equations. Compared with pure gridless method, the efficiency of the hybrid algorithm is improved to the level of finite volume method for most parts of the flow filed arc covered with grid cells. Moreover, the hybrid method is flexible to deal with the configurations as clouds of points are used to cover the region adjacent to the bodies. Mirror satellites and mirror grid cells arc introduced to the interface to accomplish data communication between the different parts of the flow field. The Euler Equations arc spatially discretized with finite volume method and gridless method in mesh and clouds of points respectively, and an explicit four-stage Runge-Kutta scheme is utilized to reach the steady-state solution. Internal flows in channels and external flows over airfoils arc investigated with hybrid method, and the solutions arc comparad to those using pure finite volume method and pure gridless method. Numerical examples show that the hybrid algorithm captures the shock waves accurately, and it is as efficient as fmite volume method.展开更多
The main purpose of this paper is to develop a gridless method for unsteady flow simulation. A quadrantal point infilling strategy is developed to generate point and combine clouds of points automatically. A point-mov...The main purpose of this paper is to develop a gridless method for unsteady flow simulation. A quadrantal point infilling strategy is developed to generate point and combine clouds of points automatically. A point-moving algorithm is introduced to ensure the clouds of points following the movements of bodyboundaries. A dual time method for solving the two-dimenslonal Euler equations in Arbitrary Lagrangian-Eulerian （ALE） formulation is presented. Dual time method allows the real-time step to be chosen on the basis of accuracy rather than stability. It also permits the acceleration techniques, which are commonly used to speed up steady flow calculations, to be used when marching the equations in pseudo time. The spatial derivatives, which are used to estimating the inviscid flux, are directly approximated by using local least-squares curve method. An explicit multistage Runge-Kutta algorithm is used to advance the flow equations in pseudo time. In order to accelerate the solution to convergence, local time stepping technique and residual averaging are employed. The results of NACA0012 airfoil in transonic steady flow are presented to verify the accuracy of the present spatial discretization method. Finally, two AGARD standard test cases in which NACA0012 airfoil and NACA64A010 airfoil oscillate in transonic flow are simulated. The computational results are compared with the experimental data to demonstrate the validity and practicality of the presented method.展开更多
A quadtree-based adaptive Cartesian grid generator and flow solver were developed. The grid adaptation based on pressure or density gradient was performed and a gridless method based on the least-square fashion was us...A quadtree-based adaptive Cartesian grid generator and flow solver were developed. The grid adaptation based on pressure or density gradient was performed and a gridless method based on the least-square fashion was used to treat the wall surface boundary condition, which is generally difficult to be handled for the common Cartesian grid. First, to validate the technique of grid adaptation, the benchmarks over a forward-facing step and double Mach reflection were computed. Second, the flows over the NACA 0012 airfoil and a two-element airfoil were calculated to validate the developed gridless method. The computational results indi- cate the developed method is reasonable for complex flows.展开更多
A hybrid Cartesian grid/gridless method is developed for calculating viscous flows over multi-element airfoils.The method adopts an unstructured Cartesian grid to cover most areas of the computational domain and leave...A hybrid Cartesian grid/gridless method is developed for calculating viscous flows over multi-element airfoils.The method adopts an unstructured Cartesian grid to cover most areas of the computational domain and leaves only small region adjacent to the aerodynamic bodies to be filled with the cloud of points used in the gridless methods,which results in a better combination of the computational efficiency of the Cartesian grid and the flexibility of the gridless method in handling complex geometries.The clouds of points in the local gridless region are implemented in an anisotropic way according to the features of the thin boundary layer of the viscous flows over the airfoils,and the clouds of points at the vicinity of the interface between the grid and the gridless regions are also controlled by using an adaptive refinement technique during the generation of the unstructured Cartesian grid.An implementation of the resulting hybrid method is presented for solving two-dimensional compressible Navier-Stokes(NS)equations.The simulations of the viscous flows over a RAE2822airfoil or a two-element airfoil are successfully carried out,and the obtained results agree well with the available experimental data.展开更多
A preconditioned gridless method is developed for solving the Euler equations at low Mach numbers.The preconditioned system in a conservation form is obtained by multiplying apreconditioning matrix of the type of Weis...A preconditioned gridless method is developed for solving the Euler equations at low Mach numbers.The preconditioned system in a conservation form is obtained by multiplying apreconditioning matrix of the type of Weiss and Smith to the time derivative of the Euler equations,which are discretized using agridless technique wherein the physical domain is distributed by clouds of points.The implementation of the preconditioned gridless method is mainly based on the frame of the traditional gridless method without preconditioning,which may fail to converge for low Mach number simulations.Therefore,the modifications corresponding to the affected terms of preconditioning are mainly addressed.The numerical results show that the preconditioned gridless method still functions for compressible transonic flow simulations and additionally,for nearly incompressible flow simulations at low Mach numbers as well.The paper ends with the nearly incompressible flow over a multi-element airfoil,which demonstrates the ability of the method presented for treating flows over complicated geometries.展开更多
The least-square gridless method was extended to simulate the compressible multi-material flows. The algorithm was accomplished to solve the Arbitrary Lagrange-Euler( ALE) formulation. The local least-square curve fit...The least-square gridless method was extended to simulate the compressible multi-material flows. The algorithm was accomplished to solve the Arbitrary Lagrange-Euler( ALE) formulation. The local least-square curve fits was adopted to approximate the spatial derivatives of a point on the base of the points in its circular support domain,and the basis function was linear. The HLLC( Harten-Lax-van Leer-Contact) scheme was used to calculate the inviscid flux. On the material interfaces,the gridless points were endued with a dual definition corresponding to different materials. The moving velocity of the interface points was updated by solving the Riemann problem. The interface boundary condition was built by using the Ghost Fluid Method( GFM).Computations were performed for several one and two dimensional typical examples. The numerical results show that the interface and the shock wave are well captured,which proves the effectiveness of gridless method in dealing with multi-material flow problems.展开更多
Efficient estimation of line spectral from quantized samples is of significant importance in information theory and signal processing,e.g.,channel estimation in energy efficient massive MIMO systems and direction of a...Efficient estimation of line spectral from quantized samples is of significant importance in information theory and signal processing,e.g.,channel estimation in energy efficient massive MIMO systems and direction of arrival estimation.The goal of this paper is to recover the line spectral as well as its corresponding parameters including the model order,frequencies and amplitudes from heavily quantized samples.To this end,we propose an efficient gridless Bayesian algorithm named VALSE-EP,which is a combination of the high resolution and low complexity gridless variational line spectral estimation(VALSE)and expectation propagation(EP).The basic idea of VALSE-EP is to iteratively approximate the challenging quantized model of line spectral estimation as a sequence of simple pseudo unquantized models,where VALSE is applied.Moreover,to obtain a benchmark of the performance of the proposed algorithm,the Cram′er Rao bound(CRB)is derived.Finally,numerical experiments on both synthetic and real data are performed,demonstrating the near CRB performance of the proposed VALSE-EP for line spectral estimation from quantized samples.展开更多
The gridless method coupled with finite rate chemistry model is employed to simulate the external combustion flow fields of M864 base bleed projectile. The fluid dynamics process is described by Euler Equation in 2-D ...The gridless method coupled with finite rate chemistry model is employed to simulate the external combustion flow fields of M864 base bleed projectile. The fluid dynamics process is described by Euler Equation in 2-D axisymmetric coordinate. The numerical method is based on least-square gridless method,and the inviscid flux is calculated by multi-component HLLC( Harten-Lax-van Leer-Contact) scheme,and a H2-CO reaction mechanism involving 9 species and 11 reactions is used. The computations are performed for the full projectile configuration of Ma = 1. 5,2,and 3. The hot air injection cases and inert cases are simulated for comparison. The numerical results show that due to the combustion in the weak region,the recirculation zone enlarges and moves downstream,the base pressure increases and the total drag force coefficient decreases. At Ma = 3. 0,the rear stagnation point shifts downstream approximate 0. 26 caliber,and the base pressure increases about 53. 4%,and the total drag force coefficient decreases to 0. 182 which agrees well with the trajectory model prediction. Due to neglecting the effects of viscosity and turbulence,there exists a certain difference at Ma = 1. 5,2. 0.展开更多
Particle swarm optimization algorithm is presented for the layout of ＂Integrate Circuit （IC）＂ design. Particle swarm optimization based on swarm intelligence is a new evolutionary computational tool and is success...Particle swarm optimization algorithm is presented for the layout of ＂Integrate Circuit （IC）＂ design. Particle swarm optimization based on swarm intelligence is a new evolutionary computational tool and is successfully applied in function optimization, neural network design, classification, pattern recognition, signal processing and robot technology and so on. A modified algorithm is presented and applied to the layout of IC design. For a given layout plane, first of all, this algorithm generates the corresponding grid group by barriers and nets＇ ports with the thought ofgridless net routing, establishes initialization fuzzy matrix, then utilizes the global optimization character to find out the best layout route only if it exits. The results of model simulation indicate that PSO algorithm is feasible and efficient in IC layout design.展开更多
A new gridless router to improve the yield of IC layout is presented. The improvement of yield is achieved by reducing the critical areas where the circuit failures are likely to happen. This gridless area router bene...A new gridless router to improve the yield of IC layout is presented. The improvement of yield is achieved by reducing the critical areas where the circuit failures are likely to happen. This gridless area router benefits from a novel cost function to compute critical areas during routing process, and heuristically lays the patterns on the chip area where it is less possible to induce critical area. The router also takes other objectives into consideration, such as routing completion rate and nets length. It takes advantage of gridless routing to gain more flexibility and a higher completion rate. The experimental results show that critical areas are effectively decreased by 21% on average while maintaining the routing completion rate over 99%.展开更多
In this paper,preconditioned gridless methods are developed for solving the threedimensional(3D)Euler equations at low Mach numbers.The preconditioned system is obtained by multiplying a preconditioning matrix of the ...In this paper,preconditioned gridless methods are developed for solving the threedimensional(3D)Euler equations at low Mach numbers.The preconditioned system is obtained by multiplying a preconditioning matrix of the type of Weiss and Smith to the time derivative of the 3D Euler equations,which are discretized under the clouds of points distributed in the computational domain by using a gridless technique.The implementations of the preconditioned gridless methods are mainly based on the frame of the traditional gridless method without preconditioning,which may fail to have convergence for flow simulations at low Mach numbers,therefore the modifications corresponding to the affected terms of preconditioning are mainly addressed in the paper.An explicit four-stage Runge–Kutta scheme is first applied for time integration,and the lower-upper symmetric Gauss-Seidel(LU-SGS)algorithm is then introduced to form the implicit counterpart to have the further speed up of the convergence.Both the resulting explicit and implicit preconditioned gridless methods are validated by simulating flows over two academic bodies like sphere or hemispherical headform,and transonic and nearly incompressible flows over one aerodynamic ONERA M6 wing.The gridless clouds of both regular and irregular points are used in the simulations,which demonstrates the ability of the method presented for coping with flows over complicated aerodynamic geometries.Numerical results of surface pressure distributions agree well with available experimental data or simulated solutions in the literature.The numerical results also show that the preconditioned gridless methods presented still functions for compressible transonic flow simulations and additionally,for nearly incompressible flow simulations at low Mach numbers as well.The convergence of the implicit preconditioned gridless method,as expected,is much faster than its explicit counterpart.展开更多
文摘A multi layer gridless area router is reported.Based on corner stitching,this router adopts tile expansion to explore path for each net.A heuristic method that penalizes nodes deviating from the destination is devised to accelerate the algorithm.Besides,an enhanced interval tree is used to manage the intermediate data structure.In order to improve the completion rate of routing,a new gridless rip up and rerouting algorithm is proposed.The experimental results indicate that the completion rate is improved after the rip up and reroute process and the speed of this algorithm is satisfactory.
基金Supported by the National Natural Science Foundation of China(10372043,11172134)the Fundingof Jiangsu Innovation Program for Graduate Education(CXZZ11-0192)~~
文摘Gridless method is developed for unsteady viscous flows involving moving boundaries. The point distri- bution of gridless method is implemented in an isotropic or anisotropic way according to the features of viscous flows. In the area far away from the body, the traditional cloud of isotropic points is used, while in the adjacent area, the cloud of anisotropic points is distributed. In this way, the point spacing normal to the wall can be small enough for simulating the boundary layer, and meanwhile, the total number of points in the computational do- main can be controlled due to large spacing in other tangential direction through the anisotropic way. A fast mov- ing technique of clouds of points at each time-step is presented based on the attenuation law of disturbed motion for unsteady flows involving moving boundaries. In the mentioned cloud of points, a uniform weighted least- square curve fit method is utilized to discretize the spatial derivatives of the Navier-Stokes equations. The pro- posed gridless method, coupled with a dual time-stepping method and the Spalart-Allmaras turbulence model, is implemented for the Navier-Stokes equations. The computational results of unsteady viscous flows around a NLR7301 airfoil with an oscillating flap and a pitching NACA0012 airfoil are presented in a good agreement with the available experimental data.
基金Aeronautical Science Foundation of China (02A52002), National Natural Science Foundation of China(10372043)
文摘A fast hybrid algorithm based on gridless method coupled with finite volume method （FVM） is developed for the solution to Euler equations. Compared with pure gridless method, the efficiency of the hybrid algorithm is improved to the level of finite volume method for most parts of the flow filed arc covered with grid cells. Moreover, the hybrid method is flexible to deal with the configurations as clouds of points are used to cover the region adjacent to the bodies. Mirror satellites and mirror grid cells arc introduced to the interface to accomplish data communication between the different parts of the flow field. The Euler Equations arc spatially discretized with finite volume method and gridless method in mesh and clouds of points respectively, and an explicit four-stage Runge-Kutta scheme is utilized to reach the steady-state solution. Internal flows in channels and external flows over airfoils arc investigated with hybrid method, and the solutions arc comparad to those using pure finite volume method and pure gridless method. Numerical examples show that the hybrid algorithm captures the shock waves accurately, and it is as efficient as fmite volume method.
文摘The main purpose of this paper is to develop a gridless method for unsteady flow simulation. A quadrantal point infilling strategy is developed to generate point and combine clouds of points automatically. A point-moving algorithm is introduced to ensure the clouds of points following the movements of bodyboundaries. A dual time method for solving the two-dimenslonal Euler equations in Arbitrary Lagrangian-Eulerian （ALE） formulation is presented. Dual time method allows the real-time step to be chosen on the basis of accuracy rather than stability. It also permits the acceleration techniques, which are commonly used to speed up steady flow calculations, to be used when marching the equations in pseudo time. The spatial derivatives, which are used to estimating the inviscid flux, are directly approximated by using local least-squares curve method. An explicit multistage Runge-Kutta algorithm is used to advance the flow equations in pseudo time. In order to accelerate the solution to convergence, local time stepping technique and residual averaging are employed. The results of NACA0012 airfoil in transonic steady flow are presented to verify the accuracy of the present spatial discretization method. Finally, two AGARD standard test cases in which NACA0012 airfoil and NACA64A010 airfoil oscillate in transonic flow are simulated. The computational results are compared with the experimental data to demonstrate the validity and practicality of the presented method.
基金supported by the National Natural Science Foundation of China (10672168)
文摘A quadtree-based adaptive Cartesian grid generator and flow solver were developed. The grid adaptation based on pressure or density gradient was performed and a gridless method based on the least-square fashion was used to treat the wall surface boundary condition, which is generally difficult to be handled for the common Cartesian grid. First, to validate the technique of grid adaptation, the benchmarks over a forward-facing step and double Mach reflection were computed. Second, the flows over the NACA 0012 airfoil and a two-element airfoil were calculated to validate the developed gridless method. The computational results indi- cate the developed method is reasonable for complex flows.
基金Supported by the National Natural Science Foundation of China(11172134)the Funding of Jiangsu Innovation Program for Graduate Education(CXZZ110192)the Priority Academic Program Development of Jiangsu Higher Education Institutions
文摘A hybrid Cartesian grid/gridless method is developed for calculating viscous flows over multi-element airfoils.The method adopts an unstructured Cartesian grid to cover most areas of the computational domain and leaves only small region adjacent to the aerodynamic bodies to be filled with the cloud of points used in the gridless methods,which results in a better combination of the computational efficiency of the Cartesian grid and the flexibility of the gridless method in handling complex geometries.The clouds of points in the local gridless region are implemented in an anisotropic way according to the features of the thin boundary layer of the viscous flows over the airfoils,and the clouds of points at the vicinity of the interface between the grid and the gridless regions are also controlled by using an adaptive refinement technique during the generation of the unstructured Cartesian grid.An implementation of the resulting hybrid method is presented for solving two-dimensional compressible Navier-Stokes(NS)equations.The simulations of the viscous flows over a RAE2822airfoil or a two-element airfoil are successfully carried out,and the obtained results agree well with the available experimental data.
基金supported by the National Natural Science Foundation of China(No.11172134)
文摘A preconditioned gridless method is developed for solving the Euler equations at low Mach numbers.The preconditioned system in a conservation form is obtained by multiplying apreconditioning matrix of the type of Weiss and Smith to the time derivative of the Euler equations,which are discretized using agridless technique wherein the physical domain is distributed by clouds of points.The implementation of the preconditioned gridless method is mainly based on the frame of the traditional gridless method without preconditioning,which may fail to converge for low Mach number simulations.Therefore,the modifications corresponding to the affected terms of preconditioning are mainly addressed.The numerical results show that the preconditioned gridless method still functions for compressible transonic flow simulations and additionally,for nearly incompressible flow simulations at low Mach numbers as well.The paper ends with the nearly incompressible flow over a multi-element airfoil,which demonstrates the ability of the method presented for treating flows over complicated geometries.
文摘The least-square gridless method was extended to simulate the compressible multi-material flows. The algorithm was accomplished to solve the Arbitrary Lagrange-Euler( ALE) formulation. The local least-square curve fits was adopted to approximate the spatial derivatives of a point on the base of the points in its circular support domain,and the basis function was linear. The HLLC( Harten-Lax-van Leer-Contact) scheme was used to calculate the inviscid flux. On the material interfaces,the gridless points were endued with a dual definition corresponding to different materials. The moving velocity of the interface points was updated by solving the Riemann problem. The interface boundary condition was built by using the Ghost Fluid Method( GFM).Computations were performed for several one and two dimensional typical examples. The numerical results show that the interface and the shock wave are well captured,which proves the effectiveness of gridless method in dealing with multi-material flow problems.
基金supported by National Natural Science Foundation of China(No.61901415)。
文摘Efficient estimation of line spectral from quantized samples is of significant importance in information theory and signal processing,e.g.,channel estimation in energy efficient massive MIMO systems and direction of arrival estimation.The goal of this paper is to recover the line spectral as well as its corresponding parameters including the model order,frequencies and amplitudes from heavily quantized samples.To this end,we propose an efficient gridless Bayesian algorithm named VALSE-EP,which is a combination of the high resolution and low complexity gridless variational line spectral estimation(VALSE)and expectation propagation(EP).The basic idea of VALSE-EP is to iteratively approximate the challenging quantized model of line spectral estimation as a sequence of simple pseudo unquantized models,where VALSE is applied.Moreover,to obtain a benchmark of the performance of the proposed algorithm,the Cram′er Rao bound(CRB)is derived.Finally,numerical experiments on both synthetic and real data are performed,demonstrating the near CRB performance of the proposed VALSE-EP for line spectral estimation from quantized samples.
文摘The gridless method coupled with finite rate chemistry model is employed to simulate the external combustion flow fields of M864 base bleed projectile. The fluid dynamics process is described by Euler Equation in 2-D axisymmetric coordinate. The numerical method is based on least-square gridless method,and the inviscid flux is calculated by multi-component HLLC( Harten-Lax-van Leer-Contact) scheme,and a H2-CO reaction mechanism involving 9 species and 11 reactions is used. The computations are performed for the full projectile configuration of Ma = 1. 5,2,and 3. The hot air injection cases and inert cases are simulated for comparison. The numerical results show that due to the combustion in the weak region,the recirculation zone enlarges and moves downstream,the base pressure increases and the total drag force coefficient decreases. At Ma = 3. 0,the rear stagnation point shifts downstream approximate 0. 26 caliber,and the base pressure increases about 53. 4%,and the total drag force coefficient decreases to 0. 182 which agrees well with the trajectory model prediction. Due to neglecting the effects of viscosity and turbulence,there exists a certain difference at Ma = 1. 5,2. 0.
文摘Particle swarm optimization algorithm is presented for the layout of ＂Integrate Circuit （IC）＂ design. Particle swarm optimization based on swarm intelligence is a new evolutionary computational tool and is successfully applied in function optimization, neural network design, classification, pattern recognition, signal processing and robot technology and so on. A modified algorithm is presented and applied to the layout of IC design. For a given layout plane, first of all, this algorithm generates the corresponding grid group by barriers and nets＇ ports with the thought ofgridless net routing, establishes initialization fuzzy matrix, then utilizes the global optimization character to find out the best layout route only if it exits. The results of model simulation indicate that PSO algorithm is feasible and efficient in IC layout design.
基金Supported by the National Natural Science Foundation of China(NSFC)under Grant No.60476014.
文摘A new gridless router to improve the yield of IC layout is presented. The improvement of yield is achieved by reducing the critical areas where the circuit failures are likely to happen. This gridless area router benefits from a novel cost function to compute critical areas during routing process, and heuristically lays the patterns on the chip area where it is less possible to induce critical area. The router also takes other objectives into consideration, such as routing completion rate and nets length. It takes advantage of gridless routing to gain more flexibility and a higher completion rate. The experimental results show that critical areas are effectively decreased by 21% on average while maintaining the routing completion rate over 99%.
基金This work was supported in part by National Natural Science Foundation of China(No.11972189)Natural Science Foundation of Jiangsu Province(No.BK20190391).
文摘In this paper,preconditioned gridless methods are developed for solving the threedimensional(3D)Euler equations at low Mach numbers.The preconditioned system is obtained by multiplying a preconditioning matrix of the type of Weiss and Smith to the time derivative of the 3D Euler equations,which are discretized under the clouds of points distributed in the computational domain by using a gridless technique.The implementations of the preconditioned gridless methods are mainly based on the frame of the traditional gridless method without preconditioning,which may fail to have convergence for flow simulations at low Mach numbers,therefore the modifications corresponding to the affected terms of preconditioning are mainly addressed in the paper.An explicit four-stage Runge–Kutta scheme is first applied for time integration,and the lower-upper symmetric Gauss-Seidel(LU-SGS)algorithm is then introduced to form the implicit counterpart to have the further speed up of the convergence.Both the resulting explicit and implicit preconditioned gridless methods are validated by simulating flows over two academic bodies like sphere or hemispherical headform,and transonic and nearly incompressible flows over one aerodynamic ONERA M6 wing.The gridless clouds of both regular and irregular points are used in the simulations,which demonstrates the ability of the method presented for coping with flows over complicated aerodynamic geometries.Numerical results of surface pressure distributions agree well with available experimental data or simulated solutions in the literature.The numerical results also show that the preconditioned gridless methods presented still functions for compressible transonic flow simulations and additionally,for nearly incompressible flow simulations at low Mach numbers as well.The convergence of the implicit preconditioned gridless method,as expected,is much faster than its explicit counterpart.
