Primal Equations Research Articles

An MP-DWR method for h-adaptive finite element methods

In a dual-weighted residual method based on the finite element framework, the Galerkin orthogonality is an issue that prevents solving the dual equation in the same space as the one for the primal equation. In the literature, there have been two popular approaches to constructing a new space for the dual problem, i.e., refining mesh grids (h-approach) and raising the order of approximate polynomials (p-approach). In this paper, a novel approach is proposed for the purpose based on the multiple-precision technique, i.e., the construction of the new finite element space is based on the same configuration as the one for the primal equation, except for the precision in calculations. The feasibility of such a new approach is discussed in detail in the paper. In numerical experiments, the proposed approach can be realized conveniently with C++ template. Moreover, the new approach shows remarkable improvements in both efficiency and storage compared with the h-approach and the p-approach. It is worth mentioning that the performance of our approach is comparable with the one through a higher order interpolation (i-approach) in the literature. The combination of these two approaches is believed to further enhance the efficiency of the dual-weighted residual method.

On-the-fly dual reduction for time-dependent topology optimization

A new model reduction approach is presented for time-dependent topology optimization of elastodynamic problems. Two salient features of the model reduction approach are on-the-fly and dual reduction. That is, the snapshots and reduced basis are obtained during the optimization rather than through off-line training; primal and adjoint equations are reduced independently. For the on-the-fly reduction, we use two complementary dynamic sampling strategies for snapshots: nearest nf full-order solutions and rolling transient ensemble of nt solutions. For the dual reduction, we apply the proper orthogonal decomposition and Galerkin procedure independently to the primal equations and to the adjoint equations for reduction. Central to our dual reduction strategy is the use of continuous adjoint to obtain the adjoint equations so that the adjoint equations can be reduced independently and along side the primal equation reduction. The residuals of the governing primal and adjoint equations are used to determine whether full-order model (FOM) based solutions should be used. Two-dimensional and three-dimensional numerical examples on topology optimization of elastodynamics problems are presented. Our numerical results demonstrate that the on-the-fly dual-reduction approach leads to two orders of magnitude improvement in FOM reduction (in terms of the number of FOM solution procedures being invoked) and data storage (in terms of storage of primal solutions required to solve the backward time-dependent adjoint equations and compute the sensitivity).

Multilevel correction goal-oriented adaptive finite element method for semilinear elliptic equations

In this study, a multilevel correction-type goal-oriented adaptive finite element method is designed for semilinear elliptic equations. Concurrently, the corresponding convergence property is theoretically proved. In the novel goal-oriented adaptive finite element method, only a linearized primal equation and a linearized dual equation are required to be solved in each adaptive finite element space. To ensure convergence, the approximate solution of the primal equation was corrected by solving a small-scale semilinear elliptic equation after the central solving process in each adaptive finite element space. Since solving of the large-scale semilinear elliptic equations is avoided and the goal-oriented technique is absorbed, there has been a significant improvement in the solving efficiency for the goal functional of semilinear elliptic equations.

Morphophoric POVMs, generalised qplexes, and 2-designs

We study the class of quantum measurements with the property that the image of the set of quantum states under the measurement map transforming states into probability distributions is similar to this set and call such measurements morphophoric. This leads to the generalisation of the notion of a qplex, where SIC-POVMs are replaced by the elements of the much larger class of morphophoric POVMs, containing in particular 2-design (rank-1 and equal-trace) POVMs. The intrinsic geometry of a generalised qplex is the same as that of the set of quantum states, so we explore its external geometry, investigating, inter alia, the algebraic and geometric form of the inner (basis) and the outer (primal) polytopes between which the generalised qplex is sandwiched. In particular, we examine generalised qplexes generated by MUB-like 2-design POVMs utilising their graph-theoretical properties. Moreover, we show how to extend the primal equation of QBism designed for SIC-POVMs to the morphophoric case.

A Newton multigrid framework for optimal control of fluid\u2013structure interactions

In this paper we consider optimal control of nonlinear time-dependent fluid structure interactions. To determine a time-dependent control variable a BFGS algorithm is used, whereby gradient information is computed via a dual problem. To solve the resulting ill conditioned linear problems occurring in every time step of state and dual equation, we develop a highly efficient monolithic solver that is based on an approximated Newton scheme for the primal equation and a preconditioned Richardson iteration for the dual problem. The performance of the presented algorithms is tested for one 2d and one 3d example numerically.

Using a simple equation to predict the microporation-enhanced transdermal drug flux

The mathematical model describing drug flux through microporated skin was previously developed. Based on this model, two mathematical equations can be used to predict the microporatio-enhanced transdermal drug flux: the complex primal equation containing a variety of experimentally-determined variables, and the simplified straightforward equation. In this study, experimental transdermal fluxes of three corticosteroids through split-thickness human skin treated with a microneedle roller were measured, and the values of fluxes compared with those predicted using both the more complex and simplified equations. According to the results of the study, both equations demonstrated high accuracy in the prediction of the fluxes of corticosteroids. The simplified equation was validated and confirmed as robust using regression analysis of literature data. Further, its capability and ease of use was exemplified by predicting the flux of methotrexate through the skin microporated with laser and comparing with published experimental data.

Adjoint analysis of Buckley-Leverett and two-phase flow equations

This paper analyzes the adjoint equations and boundary conditions for porous media flow models, specifically the Buckley-Leverett equation, and the compressible two-phase flow equations in mass conservation form. An adjoint analysis of a general scalar hyperbolic conservation law whose primal solutions include a shock jump is initially presented, and the results are later specialized to the Buckley-Leverett equation. The non-convexity of the Buckley-Leverett flux function results in adjoint characteristics that are parallel to the shock front upstream of the shock and emerge from the shock front downstream of the shock. Thus, in contrast to the behavior of Burgers’ equation where the adjoint is continuous at a shock, the Buckley-Leverett adjoint, in general, contains a discontinuous jump across the shock. Discrete adjoint solutions from space-time discontinuous Galerkin finite element approximations of the Buckley-Leverett equation are shown to be consistent with the derived closed-form analytical solutions. Furthermore, a general result relating the adjoint equations for different (though equivalent) primal equations is used to relate the two-phase flow adjoints to the Buckley-Leverett adjoint. Adjoint solutions from space-time discontinuous Galerkin finite element approximations of the two-phase flow equations are observed to obey this relationship.

A Symmetric Direct Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations

AbstractIn this work, we investigate the numerical approximation of the compressible Navier-Stokes equations under the framework of discontinuous Galerkin methods. For discretization of the viscous and heat fluxes, we extend and apply the symmetric direct discontinuous Galerkin (SDDG) method which is originally introduced for scalar diffusion problems. The original compressible Navier-Stokes equations are rewritten into an equivalent form via homogeneity tensors. Then, the numerical diffusive fluxes are constructed from the weak formulation of primal equations directly without converting the second-order equations to a first-order system. Additional numerical flux functions involving the jump of second order derivative of test functions are added to the original direct discontinuous Galerkin (DDG) discretization. A number of numerical tests are carried out to assess the practical performance of the SDDG method for the two dimensional compressible Navier-Stokes equations. These numerical results obtained demonstrate that the SDDG method can achieve the optimal order of accuracy. Especially, compared with the well-established symmetric interior penalty (SIP) method [18], the SDDG method can maintain the expected optimal order of convergence with a smaller penalty coefficient.

A hybridizable direct discontinuous Galerkin method for elliptic problems

The aim of this work is to develop a hybridizable discontinuous Galerkin method for elliptic problems. In the proposed method, the numerical flux functions are constructed from the weak formulation of primal equation directly without converting the second-order equation to a first-order system. In order to guarantee the stability and convergence of the method, we derive a computable lower bound for the constant in numerical flux functions. We also establish a prior error estimation and give some theoretical analysis for the proposed method. Finally, a numerical experiment is presented to verify the theoretical results.

Continuous adjoint-based error estimation and its application to adaptive discontinuous Galerkin method

Abstract An adaptive mesh refinement algorithm based on a continuous adjoint approach is developed. Both the primal equation and the adjoint equation are approximated with the discontinuous Galerkin (DG) method. The proposed adaptive algorithm is used in compressible Euler equations. Numerical tests are made to show the superiority of the proposed adaptive algorithm.

二重動吸振器の最大振幅倍率最小化設計

The optimal design method of double-mass dynamic vibration absorbers (DVAs) is discussed with respect to the minimization of the maximum amplitude magnification factor. The performance of a double-mass DVA is superior to a single-mass DVA with the same mass ratio, although the design methods are still the subject of studies. The design optimizations of double-mass DVAs that are arranged in parallel or series are formulated precisely using an optimality criteria approach in which the optimal parameters are obtained as the numerical solutions of simultaneous algebraic equations. The primal equations are derived using Vieta's formula with the assumption that the optimal design is realized with three resonant points of equal height. The additional equations are derived as the determinants of Jacobian matrices that are defined using the primal equations. After rearrangement, these formulations realize the direct numerical solution of the design optimization via the solution of simultaneous algebraic equations. Examples are provided that prove the effects of double-mass DVAs. The formulations used in this study are variants of the algebraic approach developed by O. Nishihara that realized the closed-form algebraic exact solutions to a popular design optimization of a single-mass DVA. The well-known design formulae that use the fixed points by J. P. Den Hartog and J. E. Brock correspond to an approximate solution to this problem.

Optimal control of the bidomain system (III): Existence of minimizers and first-order optimality conditions

We consider optimal control problems for the bidomain equations of cardiac electrophysiology together with two-variable ionic models, e.g. the Rogers–McCulloch model. After ensuring the existence of global minimizers, we provide a rigorous proof for the system of first-order necessary optimality conditions. The proof is based on a stability estimate for the primal equations and an existence theorem for weak solutions of the adjoint system.

Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations

This article presents a complete discretization of a nonlinear Sobolev equation using space-time discontinuous Galerkin method that is discontinuous in time and continuous in space. The scheme is formulated by introducing the equivalent integral equation of the primal equation. The proposed scheme does not explicitly include the jump terms in time, which represent the discontinuity characteristics of approximate solution. And then the complexity of the theoretical analysis is reduced. The existence and uniqueness of the approximate solution and the stability of the scheme are proved. The optimalorder error estimates in L2(H1) and L2(L2) norms are derived. These estimates are valid under weak restrictions on the space-time mesh, namely, without the condition kn ⩾ ch2, which is necessary in traditional space-time discontinuous Galerkin methods. Numerical experiments are presented to verify the theoretical results.

On the impact of boundary conditions on dual consistent finite difference discretizations

In this paper we derive well-posed boundary conditions for a linear incompletely parabolic system of equations, which can be viewed as a model problem for the compressible Navier–Stokes equations. We show a general procedure for the construction of the boundary conditions such that both the primal and dual equations are well-posed. The form of the boundary conditions is chosen such that reduction to first order form with its complications can be avoided.The primal equation is discretized using finite difference operators on summation-by-parts form with weak boundary conditions. It is shown that the discretization can be made energy stable, and that energy stability is sufficient for dual consistency. Since reduction to first order form can be avoided, the discretization is significantly simpler compared to a discretization using Dirichlet boundary conditions.We compare the new boundary conditions with standard Dirichlet boundary conditions in terms of rate of convergence, errors and discrete spectra. It is shown that the scheme with the new boundary conditions is not only far simpler, but also has smaller errors, error bounded properties, and highly optimizable eigenvalues, while maintaining all desirable properties of a dual consistent discretization.

Derivation of an adjoint high‐Reynolds number model for thermally driven flows

AbstractThis paper presents a high‐Reynolds number (high‐Re) approach to the adjoint Navier‐Stokes‐Fourier equations based upon modified high‐Re boundary conditions for buoyancy‐driven flows. For reasons of efficiency, the computational framework adheres to the continuous adjoint method using a frozen‐turbulence assumption. Opposed to the state of the art, the present model uses a high‐Re model for the primal equations and a high‐Re model for the adjoint equations. The adjoint high‐Re model can easily be implemented and be used for flow problems where wall functions are needed. The impact of this approach is an improved consistency of primal and adjoint equations which leads to stable and realistic shape optimisation results for industrial flows. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Adjoint-based fluid flow control and optimisation with lattice Boltzmann methods

A lattice Boltzmann (LB) framework to solve fluid flow control and optimisation problems numerically is presented. Problems are formulated on a mesoscopic basis. In a side condition, the dynamics of a Newtonian fluid is described by a family of simplified Boltzmann-like equations, namely BGK–Boltzmann equations, which are linked to an incompressible Navier–Stokes equation. It is proposed to solve the non-linear optimisation problem by a line search algorithm. The needed derivatives are obtained by deriving the adjoint equations, referred to as adjoint BGK–Boltzmann equations. The primal equations are discretised by standard lattice Boltzmann methods (LBM) while for the adjoint equations a novel discretisation strategy is introduced. The approach follows the main ideas behind LBM and is therefore referred to as adjoint lattice Boltzmann methods (ALBM). The corresponding algorithm retains most of the basic features of LB algorithms. In particular, it enables a highly-efficient parallel implementation and thus solving large-scale fluid flow control and optimisation problems. The overall solution strategy, the derivation of a prototype adjoint BGK–Boltzmann equation, the novel ALBM and its parallel realisation as well as its validation are discussed in detail in this article. Numerical and performance results are presented for a series of steady-state distributed control problems with up to approximately 1.6 million unknown control parameters obtained on a high performance computer with up to 256 processing units.

Reduced Basis Method and Error Estimation for Parametrized Optimal Control Problems with Control Constraints

We propose a Reduced Basis method for the solution of parametrized optimal control problems with control constraints for which we extend the method proposed in Dede, L. (SIAM J. Sci. Comput. 32:997, 2010) for the unconstrained problem. The case of a linear-quadratic optimal control problem is considered with the primal equation represented by a linear parabolic partial differential equation. The standard offline---online decomposition of the Reduced Basis method is employed with the Finite Element approximation as the "truth" one for the offline step. An error estimate is derived and an heuristic indicator is proposed to evaluate the Reduced Basis error on the optimal control problem at the online step; also, the indicator is used at the offline step in a Greedy algorithm to build the Reduced Basis space. We solve numerical tests in the two-dimensional case with applications to heat conduction and environmental optimal control problems.

Adjoint correction and bounding of error using lagrange form of truncation term

The a posteriori error evaluation based on differential approximation of a finite-difference scheme and adjoint equations is addressed. The differential approximation is composed of primal equations and a local truncation error determined by a Taylor series in Lagrange form. This approach provides the feasibility of both refining the solution and using the Holder inequality for asymptotic bounding of the remaining error.

連続体力学問題の積分方程式表現

The boundary integral equation method, and the boundary element method offer important advantages in approximate analysis of continuum mechanics over the so-called domain type numerical methods. This paper deals with a derivation of new integral equation formulations when establishing numerical solutions of many kinds of problems in continuum mechanics. The principal objective is to present a general method for developing such formulations, and to illustrate its application to a number of different areas of solid and fluid mechanics. Usually the fundamental equation in continuum mechanics is given in terms of the configuration variable u by Au=(T^*ET)u=f In the boundary element method, it is customary to derive the corresponding integral equation from the above equation by using the fundamental solution for the adjoint operator A^* of A. In this paper, we start with the following so-called primal set of canonical equation associated with the operator A instead of the primal equation : Kinematic relation Tu=v Balance equation T^*σ=f Constitutive equation Ev=σ From the above primal set, the new integral equation set in terms of not only the configuration variable u but also the variable σ is easily derived by using the general methodology presented in this paper. The presented method is directly applicable to numerous problems encountered in continuum mechanics. New integral equation sets and related fundamental solutions for problems of elastostatics and incompressible viscous fluid flow problems are presented illustratively. The simplicity and generality of the methodology proposed in this paper allow for the formulation of very general class of problems, non-linear, and time-dependent problems.