On the analytical derivation of the Pareto-optimal set with applications to structural design

The Fritz John conditions for Pareto-optimality have been set in matrix form and used for introducing a procedure for the analytical derivation of the Pareto-optimal set in the design variables domain. Subsequently, the derivation of the Pareto-optimal set in the objective functions domain can be obtained, if possible, by a proper analytical derivation. Both the objective and constraint functions are assumed to be available in analytical form and twice differentiable and convex (or pseudo-convex). The proposed procedure to find the Pareto-optimal set is relatively simple. The computation of the determinant of a matrix is required. A symbolic manipulator can be exploited. If there are two design variables and two objective functions, the Pareto-optimal set can be easily computed by applying a simple formula derived in the paper. If the number of design variables equals the number of objective functions, the Pareto-optimal set in the design variables domain can be found by computing the product of the constraint functions times the determinant of the Jacobian of the objective functions. A number of case studies have been proposed to test the effectiveness of the proposed procedure. The optimal structural design of, respectively, a pair of compressed spheres, a cantilever with rectangular cross section have been faced and solved. Additionally the test problem proposed by Fonseca and Fleming has been addressed and solved analytically. Optimization problems with low dimensionality (2 or 3 design variables and 2 objective functions, 2 or more constraints) have been easily solved. The proposed procedure can be useful in the actual engineering practice at the earliest design stage. In this case the designer can be made aware on the proper design variables setting to obtain the desired tradeoff among conflicting objective functions.     

A multiphase compressible model for the simulation of multiphase flows

A compressible model able to manage incompressible two-phase flows as well as compressible motions is proposed. After a presentation of the multiphase compressible concept, the new model and related numerical methods are detailed on fixed structured grids. The presented model is a 1-fluid model with a reformulated mass conservation equation which takes into account the effects of compressibility. The coupling between pressure and flow velocity is ensured by introducing mass conservation terms in the momentum and energy equations. The numerical model is then validated with four test cases involving the compression of an air bubble by water, the liquid injection in a closed cavity filled with air, a bubble subjected to an ultrasound field and finally the oscillations of a deformed air bubble in melted steel. The numerical results are compared with analytical results and convergence orders in space are provided.     

On the discontinuous Galerkin method for the simulation of compressible flow with wide range of Mach numbers

The paper is concerned with the numerical simulation of compressible flow with wide range of Mach numbers. We present a new technique which combines the discontinuous Galerkin space discretization, a semi-implicit time discretization and a special treatment of boundary conditions in inviscid convective terms. It is applicable to the solution of steady and unsteady compressible flow with high Mach numbers as well as low Mach number flow at incompressible limit without any modification of the Euler or Navier–Stokes equations.     

Pseudospectral simulation of compressible magnetohydrodynamic turbulence

We describe a conservative pseudospectral algorithm for the numerical simulation of fully compressible, dissipative magnetohydrodynamic turbulence in two spatial dimensions. Fourier collocation is employed in the two Cartesian spatial coordinates, with an isotropic truncation at each time-level. Time is explicitly discretized with a second-order Runge-Kutta scheme. We discuss conservation of material integrals and simulation of shocks. We also present results of several turbulence simulations. The results are compared with incompressible results computed by another algorithm. For the selective decay problem, the compressible results appear to converge to the incompressible case as β → ∞.     

Study of a Jacobian-free approach in the simulation of compressible fluid flows in porous media using a derivative-free spectral method

The development of Jacobian-free software for solving problems formulated by nonlinear partial differential equations is of increasing interest to simulate practical engineering processes. For the first time, this work uses the so-called derivative-free spectral algorithm for nonlinear equations in the simulation of flows in porous media. The model considered here is the one employed to describe the displacement of miscible compressible fluid in porous media with point sources and sinks, where the density of the fluid mixture varies exponentially with the pressure. This spectral algorithm is a modern method for solving large-scale nonlinear systems, which does not use any explicit information associated with the Jacobin matrix of the considered system, being a Jacobian-free approach. Two dimensional problems are presented, along with numerical results comparing the spectral algorithm to a well-developed Jacobian-free inexact Newton method. The results of this paper show that this modern spectral algorithm is a reliable and efficient method for simulation of compressible flows in porous media.     

A flux splitting method for the numerical simulation of one-dimensional compressible flow

Using the technique of flux vector splitting, it is shown that one-dimensional, inviscid, compressible-flow equations possess a split conservation form. Some attractive features of this form for the design of finite-difference solution schemes are discussed. Based on the split form, two solution chemes are designed. One is a first-order accurate ‘upwind’ scheme and the other is similar to the Lax-Wendroff scheme. A hybrid scheme, based on a nonlinear weighting of these two schemes, is demonstrated to yield results superior to either of the two in the solution of an ideal shock-tube problem.     

A volume-of-fluid method for simulation of compressible axisymmetric multi-material flow

A two-dimensional Eulerian hydrodynamic method for the numerical simulation of inviscid compressible axisymmetric multi-material flow in external force fields for the situation of pure fluids separated by macroscopic interfaces is presented. The method combines an implicit Lagrangian step with an explicit Eulerian advection step. Individual materials obey separate energy equations, fulfill general equations of state, and may possess different temperatures. Material volume is tracked using a piecewise linear volume-of-fluid method. An overshoot-free logically simple and economic material advection algorithm for cylinder coordinates is derived, in an algebraic formulation. New aspects arising in the case of more than two materials such as the material ordering strategy during transport are presented. One- and two-dimensional numerical examples are given.     

A parallelized eno procedure for direct numerical simulation of compressible turbulence

In this paper, a finite-difference based ENO (essentially nonoscillatory) procedure has been chosen for the direct numerical simulation (DNS) of compressible turbulence. The implementation of the ENO scheme follows the relatively efficient procedure in Shuet al. (1992), but the latter has been modified in the present paper to admit scalar conservation equations and to run on the iPSC/860 Paragon parallel supercomputer. DNS results with our procedure are in excellent agreement with pseudo-spectral and Padé approximation calculations in two and three dimensions. This is the case for a variety of initial conditions for compressible turbulence. The parallel algorithms presented are simple but quite efficient for DNS, with a speedup that approaches the theoretical value. Some of the attractive features include 1) minimum communication whereby a processor only communicates with two neighbors, 2) almost one hundred percent load balancing, 3) a checker-board approach to solve the Poisson equation reduces communication by a factor of approximately 2, and, 4) obtaining turbulence statistics is based on a global collect approach, which is implemented to ensure that a single number, rather than a large matrix of numbers, is communicated between processors. The ENO code presented in this paper should be quite useful in its own right, while the parallel implementation should allow the simulation of fairly realistic problems.     

An upwind local RBF-DQ method for simulation of inviscid compressible flows

In this paper, an upwind local radial basis function-based differential quadrature (RBF-DQ) scheme is presented for simulation of inviscid compressible flows with shock wave. RBF-DQ is a naturally mesh-free method. The scheme consists of two parts. The first part is to use the local RBF-DQ method to discretize the Euler equation in conservative, differential form on a set of scattered nodes. The second part is to apply the upwind method to evaluate the flux at the mid-point between the reference knot and its supporting knots. The proposed scheme is validated by its application to simulate the supersonic flow in a symmetric, convergent channel and the shock tube problem. The obtained numerical results agree very well with the theoretical data.     

Using cellular automata for porous media simulation

A cellular automata (CA) approach is proposed for simulating a fluid flow through porous materials with tortuous channels at pore level. The approach aims to combine CA methods both for constructing computer representation of porous material morphology and for simulating fluid flow through it. Morphology representation is obtained using CA whose evolution exhibits self-organization and results in a stable configuration. The latter is then used for Lattice Gas CA application to simulate fluid flow through a porous material specimen and compute its permeability properties. Special boundary conditions are introduced allowing for different smoothness of solid pore walls surface. The model has been tested on a small 2D fragment in a PC and then implemented to investigate a porous carbon electrode of a hydrogen fuel cell on 128 processors of a multiprocessor cluster.     

On the synchronized derivation depth of context-free grammars

We consider depth of derivations as a complexity measure for synchronized and ordinary context-free grammars. This measure differs from the earlier considered synchronization depth in that it counts the depth of the entire derivation tree. We consider (non-)existence of trade-offs when using synchronized grammars as opposed to non-synchronized grammars and establish lower bounds for certain classes of linear context-free languages.     

An analytical model for hybrid checkpointing in time warpdistributed simulation

The Time Warp distributed simulation algorithm uses checkpointing to save process states after certain event executions for later recovery at the time of a rollback. Two main techniques have been used for checkpointing: periodic state saving and incremental state saving. The former technique introduces large overheads in reconstructing a desired state by coasting forward from an earlier checkpointed state if the computational granularity is large. The latter technique also has large overheads in applications with large rollback distances. A hybrid checkpointing technique is proposed which uses both periodic and incremental state saving simultaneously in such a way that it reduces checkpointing time overheads. A detailed analytical model is developed for the hybrid technique, and comparisons are made using similar analytical models with periodic and incremental state saving techniques. Results show that when the system parameters are chosen to represent large and complex simulated systems, the hybrid approach has less checkpointing time overhead than the other two techniques     

Direct numerical simulation of compressible turbulent channel flows using the discontinuous Galerkin method

Direct numerical simulation of turbulent channel flows between isothermal walls have been carried out using discontinuous Galerkin method. Three Mach numbers are considered (0.2, 0.7, and 1.5) at a fixed Reynolds number ≈2800, based on the bulk velocity, bulk density, half channel width, and dynamic viscosity at the wall. Power law and log-law with the scaling of the mean streamwise velocity are considered to study their performance on compressible flows and their dependence on Mach numbers. It indicates that power law seems slightly better and less dependent on Mach number than the log-law in the overlap region. Mach number effects on the second-order (velocity, pressure, density, temperature, shear stress, and vorticity fluctuations) and higher-order (skewness and flatness of velocity, pressure, density, and temperature fluctuations) statistics are explored and discussed. Both inner (that is wall variables) and outer (that is global) scalings (with Mach number) are considered. It is found that for some second-order statistics (i.e. velocity, density, and temperature), the outer scaling collapses better than the inner scaling. It is also found that near-wall large-scale motions are affected by Mach number. The near-wall spanwise streak spacing increases with increasing Mach number. Iso-surfaces of the second invariant of the velocity gradient tensor are more sparsely distributed and elongated as Mach number increases, which is similar to the distribution of near-wall low speed streaks.     

On speeding up an asymptotic-analysis-based homogenisation scheme for designing gradient porous structured materials using a zoning strategy

Structural and Multidisciplinary Optimization - Surrogate modeling is commonly used to replace expensive simulations of engineering problems. Kriging is a popular surrogate for deterministic...     

On a class of operators for expert systems

We use the concept of directed algebra (closely related to De Morgan triplets) to modelize connectives in expert systems when linguistic terms are introduced. Mainly this article describes all directed algebra structures on a totally ordered finite set. © 1993 John Wiley & Sons, Inc.     

Large stencil viscous flux linearization for the simulation of 3D compressible turbulent flows with backward-Euler schemes

The purpose of this article is to study different approximate linearizations of the RANS equations viscous fluxes, for numerical simulations of compressible turbulent flows with backward-Euler schemes. The explicit convective flux under consideration is centred with artificial dissipation. The discrete viscous flux, calculated from cell-centred evaluation of the gradients, needs less computations and memory storage than other discretizations. But, in other respects, the balance of this numerical flux has a large stencil, which is not coherent with the 3-point per mesh direction stencil of classical implicit stages. Therefore 3-point and 5-point per mesh direction approximate linearizations are built from the thin layer flux formula. The stability condition of the corresponding backward-Euler schemes is given for a scalar linear equation (for the basic non-factored version of scheme and with LU-relaxation). Multigrid and monogrid computations of turbulent flow around two external configurations are performed with Wilcox’s k-ω turbulence model. The 5-point per mesh direction linearizations, coherent with the differential of the fluxes balance of thin layer approximation of explicit viscous fluxes, leads to the most efficient implicit stages.     

An immersed boundary method on Cartesian adaptive grids for the simulation of compressible flows around arbitrary geometries

In this article, we present an immersed boundary method for the simulation of compressible flows of complex geometries encountered in aerodynamics. The immersed boundary methods allow the mesh not to conform to obstacles, whose influence is taken into account by modifying the governing equations locally (either by a source term within the equation or by imposing the flow variables or fluxes locally, similarly to a boundary condition). A main feature of the approach which we propose is that it relies on structured Cartesian grids in combination with a dedicated HPC Cartesian solver, taking advantage of their low memory and CPU time requirements but also the automation of the mesh generation and adaptation. Turbulent flow simulations are performed by solving the Reynolds-averaged Navier–Stokes equations or by a Large-Eddy simulation approach, in combination with a wall function at high Reynolds number, to mitigate the cell count resulting from the isotropic nature of Cartesian cells. The objective of this paper is to demonstrate that this automatic workflow is fast and robust and enables to get quantitative aerodynamics results on geometrically complex configurations. Results obtained are in good agreement with classical body-fitted approaches but with a significant reduction of the time of the whole process, that is a day for RANS simulations, including the mesh generation.