Optimal boundary discretization by variational data assimilation

A variational data assimilation technique applied to the identification of the optimal discretization of interpolation operators and derivatives in the nodes adjacent to the boundary of the domain is discussed in frames of the linear shallow water model. The advantage of controlling the discretization of operators near the boundary rather than boundary conditions is shown. Assimilating data that have been produced by the same model on a finer grid, in a model on a coarse grid, we have shown that optimal discretization allows us to correct such errors of the numerical scheme as under‐resolved boundary layer and wrong wave velocity. Copyright © 2010 John Wiley & Sons, Ltd.     

GENERALIZED VARIATIONAL DATA ASSIMILATION METHOD AND NUMERICAL EXPERIMENT FOR NON-DIFFERENTIAL SYSTEM

The generalized variational data assimilation for non-differential dynamical systems is studied. There is no tangent linear model for non-differential systems and thus the general adjoint model can not be derived in the traditional way. The weak form of the original system was introduced, and then the generalized adjoint model was derived. The generalized variational data assimilation methods were developed for non-differential low dimensional system and non-differential high dimensional system with global and local observations. Furthermore, ideas in inverse problems are introduced to 4DVAR ( Fourdimensional variational ) of non-differential partial differential system with local observations.     

A Hybrid Monte‐Carlo sampling smoother for four‐dimensional data assimilation

This paper constructs an ensemble‐based sampling smoother for four‐dimensional data assimilation using a Hybrid/Hamiltonian Monte‐Carlo approach. The smoother samples efficiently from the posterior probability density of the solution at the initial time. Unlike the well‐known ensemble Kalman smoother, which is optimal only in the linear Gaussian case, the proposed methodology naturally accommodates non‐Gaussian errors and nonlinear model dynamics and observation operators. Unlike the four‐dimensional variational method, which only finds a mode of the posterior distribution, the smoother provides an estimate of the posterior uncertainty. One can use the ensemble mean as the minimum variance estimate of the state or can use the ensemble in conjunction with the variational approach to estimate the background errors for subsequent assimilation windows. Numerical results demonstrate the advantages of the proposed method compared to the traditional variational and ensemble‐based smoothing methods. Copyright © 2016 John Wiley & Sons, Ltd.     

A perturbation solution to a plate postbuckling problem

The modal coupling behavior of a compressed, imperfect plate is examined as a six degree of freedom structural system. A direct variational solution procedure is developed by the use of the functional for the total potential energy of the system. The non-linear algebraic equations generated are solved by use of a continued perturbation technique. The segmented deflection history is then developed for a variety of imperfection compositions and aspect ratios.     

Optimal solution error quantification in variational data assimilation involving imperfect models

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. If the model is ‘perfect,’ the optimal solution (analysis) error rises because of the presence of the input data errors (background and observation errors). Then, this error is quantified by the covariance matrix, which can be approximated by the inverse Hessian of an auxiliary control problem. If the model is not perfect, the optimal solution error includes an additional component because of the presence of the model error. In this paper, we study the influence of the model error on the optimal solution error covariance, considering strong and weak constraint data assimilation approaches. For the latter, an additional equation describing the model error dynamics is involved. Numerical experiments for the 1D Burgers equation illustrate the presented theory. Copyright © 2016 John Wiley & Sons, Ltd.     

4D Variational Data Analysis with Imperfect Model

One of the main hypothese made in variational data assimilation is to consider that the model is a strong constraint of the minimization, i.e. that the model describes exactly the behavior of the system. Obviously the hypothesis is never respected. We propose here an alternative to the 4D-Var that takes into account model errors by adding a nonphysical term into the model equation and controlling this term. A practical application is proposed on a simple case and a reduction of the size of control using preferred directions is introduced to make the method affordable for realistic applications. This revised version was published online in July 2006 with corrections to the Cover Date.     

Boundary conditions control for a shallow‐water model

A variational data assimilation technique was used to estimate optimal discretization of interpolation operators and derivatives in the nodes adjacent to the rigid boundary. Assimilation of artificially generated observational data in the shallow‐water model in a square box and assimilation of real observations in the model of the Black sea are discussed. It is shown in both experiments that controlling the discretization of operators near a rigid boundary can bring the model solution closer to observations as in the assimilation window and beyond the window. This type of control also allows to improve climatic variability of the model. Copyright © 2011 John Wiley & Sons, Ltd.     

Modeling study of a severe aerosol pollution event in December 2013over Shanghai China:An application of chemical data assimilation

This study focuses on the importance of initial conditions to air-quality predictions.We ran assimilation experiments using the WRF-Chem model and grid-point statistical interpolation(GSI),for a 9-day severe particulate matter pollution event that occurred in Shanghai in December 2013.In this application,GSI used a three-dimensional variational approach to assimilate ground-based PM2.5 observations into the chemical model,to obtain initial fields for the aerosol species.In our results,data assimilation significantly reduced the errors when compared to a simulation without assimilation,and improved forecasts of PM_(2.5)concentrations.Despite a drop in skill directly after the assimilation,a positive effect was present in forecasts for at least 12-24 h,and there was a slight improvement in the 48-h forecasts.In addition to performing well in Shanghai,the verification statistics for this assimilation experiment are encouraging for most of the surface stations in China.     

4D variational data assimilation for locally nested models: Complementary theoretical aspects and application to a 2D shallow water model

We consider the application of a four‐dimensional variational data assimilation method to a numerical model, which employs local mesh refinement to improve its solution. We focus on structured meshes where a high‐resolution grid is embedded in a coarser resolution one, which covers the entire domain. The formulation of the nested variational data assimilation algorithm was derived in a preliminary work (Int. J. Numer. Meth. Fluids 2008; under review). We are interested here in complementary theoretical aspects. We present first a model for the multi‐grid background error covariance matrix. Then, we propose a variant of our algorithms based on the addition of control variables in the inter‐grid transfers in order to allow for a reduction of the errors linked to the interactions between the grids. These formulations are illustrated and discussed in the test case experiment of a 2D shallow water model. Copyright © 2010 John Wiley & Sons, Ltd.     

Data assimilation for non-linear tidal models

A data assimilation procedure to incorporate measurements into a non-linear tidal model by using Kalman-filtering techniques is developed. The Kalman filter is based on the two-dimensional shallow water equations. To account for the inaccuracies, these equations are embedded into a stochastic environment by introducing a coloured system noise process into the momentum equations. The continuity equation is assumed to be perfect. The deterministic part of the equations is discretized using an ADI method, the stochastic part using the Euler scheme. Assuming that the system noise is less spatially variable than the underlying water wave process, this stochastic part can be approximated on a coarser grid than the grid used to approximate the deterministic part. A Chandrasekhar-type filter algorithm is employed to obtain the constant-gain extended Kalman filter for weakly non-linear systems. The capabilities of the filter are illustrated by applying it to the assimilation of water level measurements into a tidal model of the North Sea.     

Variational approach to the study of processes of geophysical hydro-thermodynamics with assimilation of observation data

This paper presents a variational approach to solving direct and inverse problems based on the joint use of mathematical models and data monitoring of processes of geophysical hydro-thermodynamics. This approach is used to solve problems related to environmental protection. A variational principle with weak constraints is formulated to account for uncertainties and errors in models and data. The inclusion of uncertainties makes it possible to develop direct non-iterative algorithms for sequential assimilation of data obtained by various observation systems. Some criteria and functions for controlling the quality of the natural environment are introduced into the modeling system to solve inverse problems of environmental risk assessment. A problem with data assimilation is considered for the Novosibirsk agglomeration.     

Ensemble data assimilation applied to an adaptive mesh ocean model

In this study, a first attempt has been made to introduce mesh adaptivity into the ensemble Kalman fiter (EnKF) method. The EnKF data assimilation system was established for an unstructured adaptive mesh ocean model (Fluidity, Imperial College London). The mesh adaptivity involved using high resolution mesh at the regions of large flow gradients and around the observation points in order to reduce the representativeness errors of the observations. The use of adaptive meshes unavoidably introduces difficulties in the implementation of EnKF. The ensembles are defined at different meshes. To overcome the difficulties, a supermesh technique is employed for generating a reference mesh. The ensembles are then interpolated from their own mesh onto the reference mesh. The performance of the new EnKF data assimilation system has been tested in the Munk gyre flow test case. The discussion of this paper will focus on (a) the development of the EnKF data assimilation system within an adaptive mesh model and (b) the advantages of mesh adaptivity in the ocean data assimilation model. Copyright © 2016 John Wiley & Sons, Ltd.     

Data assimilation of forecasted errors in hydrodynamic models using inter‐model correlations

Data‐assimilation techniques of the Kalman filter type are considered to be the state‐of‐the‐art approach for combining data information and deterministic numerical models with the objective of operational forecasting. This paper introduces, as an alternative, a faster and simpler data‐assimilation technique that exploits inter‐model correlations to distribute predicted errors. This scheme is performed in two steps: (i) prediction of the deterministic model errors at observation points using so‐called local linear models and (ii) distribution of the forecasted errors over the computational domain employing a scheme based on deterministic inter‐model correlations which describe the spatial nature of error structure. The method's advantage is that systematic error can be predicted by the error correction scheme, while the dynamics remain described by the deterministic model, which also establishes a basis for the spatial error distribution scheme. This relatively simple approach is inspired by original Kalman filter techniques but distinguishes error prediction and distribution in two different stages, hence allowing for data‐driven error forecasting and off‐line correction. In order to test the scheme's performance, a deterministic model of an artificial bay was constructed and run. The system was driven by specific forcing conditions and characterized by physical parameters that, in subsequent simulations, were deliberately manipulated to introduce errors into the model and test the scheme's capability. Copyright © 2007 John Wiley & Sons, Ltd.     

A new variational estimate for the effective response of hyperelastic composites

A variational method for predicting the effective properties of hyperelastic composites in terms of available estimates for “hyperelastic comparison composites” is proposed. In some cases this estimate can produce a lower bound on the effective energy-density function. This nonlinear-comparison variational procedure is specialized to classes of fiber and statistically isotropic composites with the aid of appropriate choices of comparison composites with neo-Hookean phases. The end results are given in terms of closed-form expressions for the effective strain energy-density functions, from which the stress-strain relations can be extracted analytically. Explicit analytical estimates for the overall responses of composites whose phases behaviors are governed by the Gent model are obtained. The results for the fiber composites are compared with corresponding finite element simulations of periodic fiber composites as well as with other available estimates. A fine agreement between the predictions obtained via the various estimates is revealed even in the limit of infinite contrast between the properties of the phases.     

Some theoretical problems on variational data assimilation

Theoretical aspects of variational data assimilation (VDA) for a simple model with both global and local observational data are discussed. For the VDA problems with global observational data, the initial conditions and parameters for the model are revisited and the model itself is modified. The estimates of both error and convergence rate are theoretically made and the validity of the method is proved. For VDA problem with local observation data, the conventional VDA method are out of use due to the ill-posedness of the problem. In order to overcome the difficulties caused by the ill-posedness, the initial conditions and parameters of the model are modified by using the improved VDA method, and the estimates of both error and convergence rate are also made. Finally, the validity of the improved VDA method is proved through theoretical analysis and illustrated with an example, and a theoretical criterion of the regularization parameters is proposed.     

On the canonical equations of Kirchhoff-Love theory of shells

The paper outlines a procedure to derive the canonical system of equations of the classical theory of thin shells using Reissner’s variational principle and partial variational principles. The Hamiltonian form of the Reissner functional is obtained using Lagrange multipliers to include the kinematical conditions that follow from the Kirchhoff-Love hypotheses. It is shown that the canonical system of equations can be represented in three different forms: one conventional form (five equilibrium equations) and two forms that are equivalent to it. This can be proved by reducing them to the same system of three equations. For problems with separable active and passive variables, partial variational principles are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 99–107, October 2007.     

Upwind finite difference method for miscible oil and water displacement problem with moving boundary values

The research of the miscible oil and water displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in the basin evolution as well as to the rational evaluation in prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. For the twodimensional bounded region, the upwind finite difference schemes are proposed. Some techniques, such as the calculus of variations, the change of variables, and the theory of a priori estimates, are used. The optimal orderl2-norm estimates are derived for the errors in the approximate solutions. The research is important both theoretically and practically for the model analysis in the field, the model numerical method, and the software development.     

Spectral Element Discretization of the Circular Driven Cavity. Part IV: The Navier-Stokes Equations

This paper deals with the spectral element discretization of the Navier-Stokes equations in a disk with discontinuous boundary data, which is known as the driven cavity problem. The numerical treatment does not involve any regularization of these data. Relying on a variational formulation in the primitive variables of velocity and pressure, we describe a discretization of these equations and derive error estimates in appropriate weighted Sobolev spaces. We propose an algorithm to solve the nonlinear discrete system and present numerical experiments to verify its efficiency.     

Numerical analysis of a quasistatic piezoelectric problem with damage

The quasistatic evolution of the mechanical state of a piezoelectric body with damage is numerically studied in this paper. Both damage and piezoelectric effects are included into the model. The variational formulation leads to a coupled system composed of two linear variational equations for the displacement field and the electric potential, and a nonlinear parabolic variational equation for the damage field. The existence of a unique weak solution is stated. Then, a fully discrete scheme is introduced by using a finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, a two-dimensional example is presented to demonstrate the behaviour of the solution. To cite this article: J.R. Fernández et al., C. R. Mecanique 336 (2008).     

A nonlocal model with strain-based damage

A thermodynamically consistent formulation of nonlocal damage in the framework of the internal variable theories of inelastic behaviours of associative type is presented. The damage behaviour is defined in the strain space and the effective stress turns out to be additively splitted in the actual stress and in the nonlocal counterpart of the relaxation stress related to damage phenomena. An important advantage of models with strain-based loading functions and explicit damage evolution laws is that the stress corresponding to a given strain can be evaluated directly without any need for solving a nonlinear system of equations. A mixed nonlocal variational formulation in the complete set of state variables is presented and is specialized to a mixed two-field variational formulation. Hence a finite element procedure for the analysis of the elastic model with nonlocal damage is established on the basis of the proposed two-field variational formulation. Two examples concerning a one-dimensional bar in simple tension and a two-dimensional notched plate are addressed. No mesh dependence or boundary effects are apparent.