In the adaptive mesh refinement technique we start with a base coarse grid. As the solution proce... more In the adaptive mesh refinement technique we start with a base coarse grid. As the solution proceeds we identify the regions requiring more resolution by some parameter characterizing the solution, say the local truncation error. We superimpose finer sub grids only on these regions. Finer and finer sub grids are added recursively until either a given maximum level of refinement is reached or the local truncation error has dropped below the desired level. Thus in an adaptive mesh refinement computation grid spacing is fixed for the base grid only and is determined locally for the sub grids according to the requirements of the problem
The immersed boundaries (IB) method allows one to greatly simplify the grid generation and even t... more The immersed boundaries (IB) method allows one to greatly simplify the grid generation and even to automate it completely. The governing equations are solved directly on a grid in their simplest form by means of very efficient numerical schemes. The grid generator detects the cell faces that are cut by the body surface and divides the cells into three types: solid and fluid cells, whose centers lie within the body and within the fluid, respectively; and fluid/solid interface cells, which have at least one of their neighbors inside the body/fluid. Then, the centers of the fluid and solid-interface cells are projected onto the body surface along its normal direction, so as to obtain fluid-cells projection points and solid-cell projection points.
Lattice Boltzmann Methods (LBM) or Thermal Lattice Boltzmann Methods (TLBM) is a CFD methods for ... more Lattice Boltzmann Methods (LBM) or Thermal Lattice Boltzmann Methods (TLBM) is a CFD methods for fluid simulation. Instead of solving the Navier–Stokes equations, the discrete Boltzmann equation is solved to simulate the flow of a Newtonian fluid with collision models such as Bhatnagar–Gross–Krook (BGK). By simulating streaming and collision processes across a limited number of particles, the intrinsic particle interactions evince a microcosm of viscous flow behavior applicable across the greater mass1. It is a modern approach in Computational Fluid Dynamics and often used to solve the incompressible, time-dependent Navier-Stokes equations numerically. Its strength lies however in the ability to easily represent complex physical phenomena, ranging from multiphase flows to chemical interactions between the fluid and the surroundings.
In mathematics and computer science, an optimization problem is the problem of finding the best s... more In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. In the simplest terms, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function.
Multigrid methods are effective techniques that can deliver fast convergence rates with minimal m... more Multigrid methods are effective techniques that can deliver fast convergence rates with minimal memory overheads. The basic idea of a multigrid method is to accelerate the solution of a set of fine grid equations by time stepping on a sequence of fine and coarse grids using a simple explicit scheme.
Anisotropic meshes are desirable for various applications, such as the numerical solving of parti... more Anisotropic meshes are desirable for various applications, such as the numerical solving of partial differential equations and graphics. From an equivalent point of view, the use of meshes whose elements are stretched according to the anisotropy requires a lower number of elements to achieve the same precision of the result of the simulation. When stretched elements are used, the mesh is said to be Anisotropic. Additionally to providing increased accuracy in the simulations of scientific modeling, anisotropic meshes also find use in geometric modeling as they can improve the visualization of objects, and lower the number of vertices required to represent a shape or interpolate a smooth function. By requiring fewer elements, anisotropic meshes thus provide another way to accelerate mesh generation, and increase the quality and the speed of computations.
Computer-Aided Design (CAD) is the use of computer systems (or workstations) to aid in the creati... more Computer-Aided Design (CAD) is the use of computer systems (or workstations) to aid in the creation, modification, analysis, or optimization of a design. CAD software is used to increase the productivity of the designer, improve the quality of design, improve communications through documentation, and to create a database for manufacturing. CAD output is often in the form of electronic files for print, machining, or other manufacturing operations. The term CADD (for Computer Aided Design and Drafting) is also used. CAD may be used to design curves and figures in two-dimensional (2D) space; or curves, surfaces, and solids in three-dimensional (3D) space. CAD is an important industrial art extensively used in many applications, including automotive, shipbuilding, and aerospace industries, industrial and architectural design, prosthetics, and many more. CAD is also widely used to produce computer animation for special effects in movies, advertising and technical manuals, often called DCC digital content creation. The modern ubiquity and power of computers means that even perfume bottles and shampoo dispensers are designed using techniques unheard of by engineers of the 1960s. Because of its enormous economic importance, CAD has been a major driving force for research in computational geometry, computer graphics (both hardware and software), and discrete differential geometry.
In the adaptive mesh refinement technique we start with a base coarse grid. As the solution proce... more In the adaptive mesh refinement technique we start with a base coarse grid. As the solution proceeds we identify the regions requiring more resolution by some parameter characterizing the solution, say the local truncation error. We superimpose finer sub grids only on these regions. Finer and finer sub grids are added recursively until either a given maximum level of refinement is reached or the local truncation error has dropped below the desired level. Thus in an adaptive mesh refinement computation grid spacing is fixed for the base grid only and is determined locally for the sub grids according to the requirements of the problem
The immersed boundaries (IB) method allows one to greatly simplify the grid generation and even t... more The immersed boundaries (IB) method allows one to greatly simplify the grid generation and even to automate it completely. The governing equations are solved directly on a grid in their simplest form by means of very efficient numerical schemes. The grid generator detects the cell faces that are cut by the body surface and divides the cells into three types: solid and fluid cells, whose centers lie within the body and within the fluid, respectively; and fluid/solid interface cells, which have at least one of their neighbors inside the body/fluid. Then, the centers of the fluid and solid-interface cells are projected onto the body surface along its normal direction, so as to obtain fluid-cells projection points and solid-cell projection points.
Lattice Boltzmann Methods (LBM) or Thermal Lattice Boltzmann Methods (TLBM) is a CFD methods for ... more Lattice Boltzmann Methods (LBM) or Thermal Lattice Boltzmann Methods (TLBM) is a CFD methods for fluid simulation. Instead of solving the Navier–Stokes equations, the discrete Boltzmann equation is solved to simulate the flow of a Newtonian fluid with collision models such as Bhatnagar–Gross–Krook (BGK). By simulating streaming and collision processes across a limited number of particles, the intrinsic particle interactions evince a microcosm of viscous flow behavior applicable across the greater mass1. It is a modern approach in Computational Fluid Dynamics and often used to solve the incompressible, time-dependent Navier-Stokes equations numerically. Its strength lies however in the ability to easily represent complex physical phenomena, ranging from multiphase flows to chemical interactions between the fluid and the surroundings.
In mathematics and computer science, an optimization problem is the problem of finding the best s... more In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. In the simplest terms, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function.
Multigrid methods are effective techniques that can deliver fast convergence rates with minimal m... more Multigrid methods are effective techniques that can deliver fast convergence rates with minimal memory overheads. The basic idea of a multigrid method is to accelerate the solution of a set of fine grid equations by time stepping on a sequence of fine and coarse grids using a simple explicit scheme.
Anisotropic meshes are desirable for various applications, such as the numerical solving of parti... more Anisotropic meshes are desirable for various applications, such as the numerical solving of partial differential equations and graphics. From an equivalent point of view, the use of meshes whose elements are stretched according to the anisotropy requires a lower number of elements to achieve the same precision of the result of the simulation. When stretched elements are used, the mesh is said to be Anisotropic. Additionally to providing increased accuracy in the simulations of scientific modeling, anisotropic meshes also find use in geometric modeling as they can improve the visualization of objects, and lower the number of vertices required to represent a shape or interpolate a smooth function. By requiring fewer elements, anisotropic meshes thus provide another way to accelerate mesh generation, and increase the quality and the speed of computations.
Computer-Aided Design (CAD) is the use of computer systems (or workstations) to aid in the creati... more Computer-Aided Design (CAD) is the use of computer systems (or workstations) to aid in the creation, modification, analysis, or optimization of a design. CAD software is used to increase the productivity of the designer, improve the quality of design, improve communications through documentation, and to create a database for manufacturing. CAD output is often in the form of electronic files for print, machining, or other manufacturing operations. The term CADD (for Computer Aided Design and Drafting) is also used. CAD may be used to design curves and figures in two-dimensional (2D) space; or curves, surfaces, and solids in three-dimensional (3D) space. CAD is an important industrial art extensively used in many applications, including automotive, shipbuilding, and aerospace industries, industrial and architectural design, prosthetics, and many more. CAD is also widely used to produce computer animation for special effects in movies, advertising and technical manuals, often called DCC digital content creation. The modern ubiquity and power of computers means that even perfume bottles and shampoo dispensers are designed using techniques unheard of by engineers of the 1960s. Because of its enormous economic importance, CAD has been a major driving force for research in computational geometry, computer graphics (both hardware and software), and discrete differential geometry.
The use of computational simulation to scan many alternative designs has proved extremely valuabl... more The use of computational simulation to scan many alternative designs has proved extremely valuable in practice, but it is also evident that the number of possible design variations is too large to permit their complete evaluation. Thus it is very unlikely that a truly optimum solution can be found without the assistance of automatic optimization procedures. As an vital “ingredients” in gradient base optimization, the sensitivities may now be estimated by making a small variation in each design parameter in turn and recalculating the flow. The gradient can be determined directly or indirectly by number of available methods, including Direct Differentiation (DD), Adjoin Variable(AV), Symbolic Differentiation (SD), Automatic Differentiation (AD), and Finite Difference (FD). Once the gradient has been calculated, a descent method can be used to determine a shape change that will make an improvement in the design.
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that use... more Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical analysis and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial experimental validation of such software is performed using a wind tunnel with the final validation coming in full-scale testing, e.g. flight tests. In design and development, CFD programs are now considered to be standard numerical tools which predict not only fluid flow behavior, but also the transfer of heat, mass (such as in perspiration or dissolution), phase change (such as in freezing, melting or boiling), chemical reaction (such as combustion or rusting), mechanical movement (such as an impeller turning, pistons, fans or rudders) and stress or deformation of related solid structures (such as a mast bending in the wind). Furthermore, CFD has been applied to deal with problems in environment and architecture.
An algorithm is developed to obtain the grid sensitivity with respect to design parameters for ae... more An algorithm is developed to obtain the grid sensitivity with respect to design parameters for aerodynamic optimization. The procedure is advocating a novel (geometrical) parameterization using spline functions such as NURBS (Non-Uniform Rational B-Splines ...
A parametric model is presented for the blendedwing -body airplane, one concept being proposed fo... more A parametric model is presented for the blendedwing -body airplane, one concept being proposed for the next generation of large subsonic transports. The model is defined in terms of a small set of parameters which facilitates analysis and optimization during the conceptual design process. The model is generated from a preliminary CAD geometry. From this geometry, airfoil cross sections are cut at selected locations and fitted with analytic curves. The airfoils are then used as boundaries for surfaces defined as the solution of partial differential equations. Both the airfoil curves and the surfaces are generated with free parameters selected to give a good representation of the original geometry. The original surface is compared with the parametric model, and solutions of the Euler equations for compressible flow are computed for both geometries. The parametric model is a good approximation of the CAD model and the computed solutions are qualitatively similar. An optimal NURBS appro...
Sensitivity analysis in Computational Fluid Dynamics with emphasis on grids and surface parameter... more Sensitivity analysis in Computational Fluid Dynamics with emphasis on grids and surface parameterization is described. An interactive algebraic grid-generation technique is employed to generate C-type grids around NACA four-digit wing sections. An analytical procedure is developed for calculating grid sensitivity with respect to design parameters of a wing section. A comparison of the sensitivity with that obtained using a finite-difference approach is made. Grid sensitivity with respect to grid parameters, such as grid-stretching coefficients, are also investigated. Using the resultant grid sensitivity, aerodynamic sensitivity is obtained using the compressible two-dimensional thin-layer Navier-Stokes equations.
A simple procedure has been developed and applied for the grid generation around an airplane geom... more A simple procedure has been developed and applied for the grid generation around an airplane geometry. This approach is based on a transfinite interpolation with Lagrangian interpolation for the blending functions. By using a Lagrangian interpolation function, it is possible to enforce the grid continuity across the block interfaces without the derivative information. Monotonic rational quadratic spline interpolation has been employed for the grid distributions on the boundaries. This allows any arbitrary grid spacing without overlapping of the grid lines. An efficient computer program has been developed to generate a multiblock grid around a generic airplane geometry. This procedure has proven to be very simple and effective.
A parametric model is presented for the blended-wing-body airplane, one concept being proposed fo... more A parametric model is presented for the blended-wing-body airplane, one concept being proposed for the next generation of large subsonic transports. The model is dened in terms of a small set of parameters which facilitates analysis and optimization during the conceptual design process. The model is generated from a preliminary CAD geometry. From this geometry, airfoil cross sections are cut at selected locations and tted with analytic curves. The airfoils are then used as boundaries for surfaces dened as the solution of partial dierential equations. Both the airfoil curves and the surfaces are generated with free parameters selected to give a good representation of the original geometry. The original surface is compared with the parametric model, and solutions of the Euler equations for compressible ow are computed for both geometries. The parametric model is a good approximation of the CAD model and the computed solutions are qualitatively similar. An optimal NURBS approximation is constructed and can be used by a CAD model for further renement or modication of the original geometry.
A parametric model is presented for the blended-wing-body airplane, one concept being proposed fo... more A parametric model is presented for the blended-wing-body airplane, one concept being proposed for the next generation of large subsonic transports. The model is dened in terms of a small set of parameters which facilitates analysis and optimization during the conceptual design process. The model is generated from a preliminary CAD geometry. From this geometry, airfoil cross sections are cut at selected locations and tted with analytic curves. The airfoils are then used as boundaries for surfaces dened as the solution of partial dierential equations. Both the airfoil curves and the surfaces are generated with free parameters selected to give a good representation of the original geometry. The original surface is compared with the parametric model, and solutions of the Euler equations for compressible ow are computed for both geometries. The parametric model is a good approximation of the CAD model and the computed solutions are qualitatively similar. An optimal NURBS approximation is constructed and can be used by a CAD model for further renement or modication of the original geometry.
An algorithm is developed to obtain the grid sensitivity with respect to design parameters for ae... more An algorithm is developed to obtain the grid sensitivity with respect to design parameters for aerodynamic optimization. The procedure is advocating a novel (geometrical) parameterization using spline functions such as NURBS (Non-Uniform Rational BSplines) for defining the airfoil geometry. An interactive algebraic grid generation technique is employed to generate C-type grids around airfoils. The grid sensitivity of the domain with respect to geometric design parameters has been obtained by direct differentiation of the grid equations. A hybrid approach is proposed for more geometrically complex configurations such as a wing or fuselage. The aerodynamic sensitivity coefficients are obtained by direct differentiation of the compressible two-dimensional thin layer Navier-Stokes equations. An optimization package has been introduced into the algorithm in order to optimize the airfoil surface.
An algorithm is developed to obtain the grid sensitivity with respect to
design parameters for ae... more An algorithm is developed to obtain the grid sensitivity with respect to design parameters for aerodynamic optimization. Two distinct parameterization procedures are developed for investigating the grid sensitivity with respect to design parameters of a wing-section as an example. The first procedure is based on traditional (physical) relations defining NACA four-digit wing-sections. The second is advocating a novel (geometrical) parameterization using spline functions such as NURBS (Non-Uniform Rational B-Splines) for defining the wing-section geometry. An interactive algebraic grid generation technique, known as Two-Boundary Grid Generation (TBGG) is employed to generate C-type grids around wing-sections. The grid sensitivity of the domain with respect to design and grid parameters has been obtained by direct differentiation of the grid equations. A hybrid approach is proposed for more geometrically complex configurations. A comparison of the sensitivity coeffÉcients with those obtained using a finite-difference approach is made to verify the feasibility of the approach. The aerodynamic sensitivity coefficients are obtained using the compressible two-dimensional thin-layer Navier-Stokes equations. An optimization package has been introduced into the algorithm in order to optimize the wing-section surface using both physical and geometric parameterization. Results demonstrate a substantially improved design, particularly in the geometric parameterization case.
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Teaching Documents by Ideen Sadrehaghighi
design parameters for aerodynamic optimization. Two distinct parameterization procedures
are developed for investigating the grid sensitivity with respect to design parameters
of a wing-section as an example. The first procedure is based on traditional
(physical) relations defining NACA four-digit wing-sections. The second is advocating
a novel (geometrical) parameterization using spline functions such as NURBS
(Non-Uniform Rational B-Splines) for defining the wing-section geometry. An interactive
algebraic grid generation technique, known as Two-Boundary Grid Generation
(TBGG) is employed to generate C-type grids around wing-sections. The grid sensitivity
of the domain with respect to design and grid parameters has been obtained by
direct differentiation of the grid equations. A hybrid approach is proposed for more
geometrically complex configurations. A comparison of the sensitivity coeffÉcients
with those obtained using a finite-difference approach is made to verify the feasibility
of the approach. The aerodynamic sensitivity coefficients are obtained using the
compressible two-dimensional thin-layer Navier-Stokes equations. An optimization
package has been introduced into the algorithm in order to optimize the wing-section
surface using both physical and geometric parameterization. Results demonstrate a
substantially improved design, particularly in the geometric parameterization case.