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• Modelling gravitational instabilities: Slab break–off and Rayleigh–Taylor diapirism

  S. Zlotnik, M. Fernández, P. Diez, J. Vergés

  Pure and Applied Geophysics (2008). Vol. 165 (8), pp. 1491-1510

  
  

  Abstract

  A non-standard new code to solve multiphase viscous thermo–mechanical problems applied to geophysics is presented. Two numerical methodologies employed in the code are described: A level set technique to track the position of the materials and an enrichment of the solution to allow the strain rate to be discontinuous across the interface. These techniques have low computational cost and can be used in standard desktop PCs. Examples of phase tracking with level set are presented in two and three dimensions to study slab detachment in subduction processes and Rayleigh–Taylor instabilities, respectively. The modelling of slab detachment processes includes realistic rheology with viscosity depending on temperature, pressure and strain rate; shear and adiabatic heating mechanisms; density including mineral phase changes and varying thermal conductivity. Detachment models show a first prolonged period of thermal diffusion until a fast necking of the subducting slab results in the break–off. The influence of several numerical and physical parameters on the detachment process is analyzed: The shear heating exerts a major influence accelerating the detachment process, reducing the onset time to one half and lubricating the sinking of the detached slab. The adiabatic heating term acts as a thermal stabilizer. If the mantle temperature follows an adiabatic gradient, neglecting this heating term must be included, otherwise all temperature contrasts are overestimated. As expected, the phase change at 410 km depth (olivine–spinel transition) facilitates the detachment process due to the increase in negative buoyancy. Finally, simple plume simulations are used to show how the presented numerical methodologies can be extended to three dimensions.

  Abstract
  A non-standard new code to solve multiphase viscous thermo–mechanical problems applied to geophysics is presented. Two numerical methodologies employed in the code are [...]

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• Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the Discontinuous Galerkin time discretization

  N. Parés, P. Diez, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (19-20), pp. 1641-1660

  
  

  Abstract

  Classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest. Consequently, the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh. Exact bounds, that is bounds guaranteed with respect to the exact solution, are needed to properly certify the accuracy of the results, especially if the meshes are coarse. The paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems. In this first part, the bounds account for the error associated with the spatial discretization. The error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error. The time discretization is performed using the discontinuous Galerkin method, both for the primal and adjoint problems. In the error estimation procedure, equilibrated fluxes at interelement edges are calculated using hybridization techniques.

  Abstract
  Classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest. Consequently, the bounds [...]

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• Bounds of functional outputs for parabolic problems. Part II: Bounds of the exact solution

  N. Parés, P. Diez, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (19-20), 1661-1679

  
  

  Abstract

  The paper introduces a methodology to compute upper and lower bounds for linear-functional outputs of the exact solutions of parabolic problems. In this second part, the bounds account for the error both in space and time. The assumption stating that the error introduced by the time marching scheme is negligible, used in the first part, is removed here. The bounds are computed starting from an approximation of the exact solution, associated with a spatial mesh and a time grid. Nevertheless, the bounds are guaranteed with respect to the exact solution, with no reference to any mesh or time discretization.

  Abstract
  The paper introduces a methodology to compute upper and lower bounds for linear-functional outputs of the exact solutions of parabolic problems. In this second part, the bounds [...]

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• Numerical modelling of tectonic plates subduction using X-FEM

  S. Zlotnik, P. Diez, M. Fernández, J. Vergés

  Comput. Methods Appl. Mech. Engrg., (2007). Vol. 196 (41-44), pp. 4283-4293

  
  

  Abstract

  The numerical modelling of plate subduction requires solving a coupled thermo-mechanical highly-nonlinear transient problem. The mechanical description of the phenomenon results in a multiphase quasi-static Stokes flow, where the inertia terms are neglected. The transient thermal problem is dominated by the advection term. Here, the representation and evolution of the different phases are described using level sets. The phase tracking is carried out transporting the level set along with the material, using a pure advective model. The gradient discontinuities induced by the viscosity jump across the interface are resolved numerically by enriching the solution using a partition of unity method in a eXtended Finite Element Method (X-FEM) context. These numerical tools are used to simulate plate subduction with different parameters and to derive useful correlations between relevant geophysical factors.

  Abstract
  The numerical modelling of plate subduction requires solving a coupled thermo-mechanical highly-nonlinear transient problem. The mechanical description of the phenomenon results [...]

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• Equilibrated Patch Recovery error estimates: simple and accurate upper bounds of the error

  P. Diez, J. Ródenas, O. Zienkiewicz

  Int. J. Numer. Meth. Engng. (2007). Vol. 69 (10), pp. 2075-2098

  
  

  Abstract

  This paper introduces a new recovery‐type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error. The upper bound property requires the recovered solution to be both statically equilibrated and continuous. The equilibrium is obtained locally (patch‐by‐patch) and the continuity is enforced by a postprocessing based on the partition of the unity concept. This postprocess is expected to preserve the features of the locally equilibrated stress field. Nevertheless, the postprocess phase modifies the equilibrium, which is no longer exactly fulfilled. A new methodology is introduced that yields upper bound estimates by taking into account this lack of equilibrium. This requires computing the ℒ︁2 norm of the error or relating it with the energy norm. The guaranteed upper bounds are obtained by using a pessimistic bound of the error ℒ︁2 norm, derived from an eigenvalue problem. Nevertheless, these bounds are not sharp. An additional strategy based on a more accurate assessment of the error ℒ︁2 norm is introduced, providing sharp estimates, which are practical upper bounds as it is demonstrated in the numerical tests.

  Abstract
  This paper introduces a new recovery‐type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error. The upper bound property requires [...]

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• Goal-oriented error estimation for transient parabolic problems

  P. Diez, G. Calderón

  Computational Mechanics (2007). Vol. 39 (5), pp. 631-646

  
  

  Abstract

  This work focuses on controlling the error and adapting the discretization in the context of parabolic problems. In order to obtain a sound mathematical framework, the time domain is discretized using a Discontinuous Galerkin (DG) approach. This allows to formulate the time stepping procedure in a variational format. The error is measured in the basis of an output of interest of the solution, defined by a linear functional. A dual problem, associated with this linear output is introduced. The dual problem has to be solved backward in time. An error representation is introduced, based on the weak residual of the primal error applied to the dual solution. Two different alternatives are studied to estimate the error in the dual solution: 1) recovery based error estimators and 2) implicit residual type estimators. Once the error assessment is performed implicitly in the dual problem, the obtained estimate is plugged into the primal residual to obtain the error in the quantity of interest. The implementation of the estimator is drastically simplified by using the weak version of the residual instead of the strong version used in previous works. Thus, the output error is assessed using a mixed technique, explicit for the primal problem and implicit for the dual. In the framework of adaptive computations of transient problems, this approach is very attractive because it allows using first the implicit scheme for the dual problem and then integrating the primal problem, estimating the error explicitly and eventually adapting the space-time grid. Thus, at every time step of the time marching scheme, the estimate of the dual error is injected into the primal residual (explicit estimate for the primal problem).

  Abstract
  This work focuses on controlling the error and adapting the discretization in the context of parabolic problems. In order to obtain a sound mathematical framework, the time [...]

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• Remeshing criteria and proper error representations for goal oriented h-adaptivity

  P. Diez, G. Calderón

  Comput. Methods Appl. Mech. Engrg., (2007). Vol. 196 (4-6), pp. 719-733

  
  

  Abstract

  A key ingredient in h-adaptivity pertains to the transformation of output data from a given error estimator into input data, usually in the form of an element-size distribution, that needs to be supplied to a mesh generator. This paper analyzes the different possibilities of de¯ning remeshing criteria in the framework of goal oriented adaptivity. In standard energy norm driven adaptivity, the optimal mesh is clearly obtained if the local error distribution is uniform. The goal-oriented paradigm introduces new di±culties associated with the different possibilities for the spatial error representation and the signs of the local error contributions. A nodal error representation is introduced in order to improve the communication with the automatic mesh generation tool, precluding the transfer of information from elements to nodes. Numerical experiments demonstrate the ability of the introduced remeshing strategies to drive e±cient adaptive procedures and to control the error in quantities of interest. The results of the numerical tests fit the expected properties of the different remeshing strategies.

  Abstract
  A key ingredient in h-adaptivity pertains to the transformation of output data from a given error estimator into input data, usually in the form of an element-size distribution, [...]

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• Subdomain-based flux-free a posteriori error estimators

  N. Parés, P. Diez, A. Huerta

  Comput. Methods Appl. Mech. Engrg., (2006). Vol. 195 (4-6), pp. 297-323

  
  

  Abstract

  A new residual type flux-free error estimator is presented. It estimates upper and lower bounds of the error in energy norm. The proposed approach precludes the main drawbacks of standard residual type estimators, circumvents the need of flux-equilibration and results in a simple implementation that uses standard resources available in .nite element codes. This is specially interesting for 3D applications where the implementation of this technique is as simple as in 2D. Recall that on the contrary, the complexity of the flux-equilibration techniques increases drastically in the 3D case. The bounds for the energy norm of the error are used to produce upper and lower bounds of linear functional outputs, representing quantities of engineering interest. The presented estimators demonstrate their efficiency in numerical tests producing sharp estimates both for the energy and the quantities of interest.

  Abstract
  A new residual type flux-free error estimator is presented. It estimates upper and lower bounds of the error in energy norm. The proposed approach precludes the main drawbacks [...]

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• Accurate upper and lower error bounds by solving flux-free local problems in “stars”

  P. Diez, N. Parés, A. Huerta

  European Journal of Finite Elements (2004). Vol. 13 (5-7), pp. 497-507

  
  

  Abstract

  Two implicit residual type estimators yielding upper bounds of the error are presented which do not require flux equilibration. One of them is based on the ideas introduced in [MAC 00, CAR 99, MOR 03, PRU 02]. The new approach introduced here is based on using the estimated error function rather than the estimated error norms. Once the upper bounds are computed, also lower bounds for the error are obtained with little supplementary effort.

  Abstract
  Two implicit residual type estimators yielding upper bounds of the error are presented which do not require flux equilibration. One of them is based on the ideas introduced [...]

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• Convergence of finite elements enriched with meshless methods

  S. Fernández-Méndez, P. Diez, A. Huerta

  Numerische Mathematik (2003). Vol. 96 (1), pp. 43-59

  
  

  Abstract

  A combined hierarchical approximation based on finite elements and mesh-less methods is proposed and studied. Finite Elements are enriched adding hierarchical shape functions based on a particle distribution. Convergence results are presented and proved.

  Abstract
  A combined hierarchical approximation based on finite elements and mesh-less methods is proposed and studied. Finite Elements are enriched adding hierarchical shape functions [...]

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