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• A simple strategy to assess the error in the numerical wave number of the finite element solution of the Helmholtz equation

  L. Steffens, P. Diez

  Comput. Methods Appl. Mech. Engrg., (2009). Vol. 198 (15-16), pp. 1389-1400

  
  

  Abstract

  The standard approach for goal oriented error estimation and adaptivity uses an error representation via an adjoint problem, based on the linear functional output representing the quantity of interest. For the assessment of the error in the approximation of the wave number for the Helmholtz problem (also referred to as dispersion or pollution error), this strategy cannot be applied. This is because there is no linear extractor producing the wave number from the solution of the acoustic problem. Moreover, in this context, the error assessment paradigm is reverted in the sense that the exact value of the wave number, κ, is known (it is part of the problem data) and the effort produced in the error assessment technique aims at obtaining the numerical wave number, κH, as a postprocess of the numerical solution, uH. The strategy introduced in this paper is based on the ideas used in the a priori analysis. A modified equation corresponding to a modified wave number κm is introduced. Then, the value of κm such that the modified problem better accommodates the numerical solution uH is taken as the estimate of the numerical wave number κH. Thus, both global and local versions of the error estimator are proposed. The obtained estimates of the dispersion error match the a priori predicted dispersion error and, in academical examples, the actual values of the error in the wave number.

  Abstract
  The standard approach for goal oriented error estimation and adaptivity uses an error representation via an adjoint problem, based on the linear functional output representing [...]

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• Bounds for quantities of interest and adaptivity in the Element Free Galerkin method

  Y. Vidal, N. Parés, P. Diez, A. Huerta

  Int. J. Numer. Meth. Engng. (2008). Vol. 76 (11), pp. 1782-1818

  
  

  Abstract

  A novel approach for implicit residual-type error estimation in mesh-free methods and an adaptive refinement strategy are presented. This allows computing upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh-free methods. The adaptive strategy proposed leads to a fast convergence of the bounds to the desired precision.

  Abstract
  A novel approach for implicit residual-type error estimation in mesh-free methods and an adaptive refinement strategy are presented. This allows computing upper and lower [...]

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• Small-scale gravitational instabilities under the oceans: implications for the evolution of oceanic lithosphere and its expression in geophysical observables

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

  Philosophical Magazine (2008). Vol. 88 (28-29), pp. 3197-3217

  
  

  Abstract

  Sublithospheric small-scale convection (SSC) is thought to be responsible for the flattening of the seafloor depth and surface heat flow observed in mature plates. Although the existence of SSC is generally accepted, its ability to effectively produce a constant lithospheric thickness (i.e. flattening of observables) is a matter of debate. Here we study the development and evolution of SSC with a 2D thermomechanical finite-element code. Emphasis is put on (i) the influence of various rheological and thermophysical parameters on SSC, and (ii) its ability to reproduce geophysical observables (i.e. seafloor depth, surface heat flow, and seismic velocities). We find that shear heating plays no significant role either in the onset of SSC or in reducing the lithospheric thickness. In contrast, radiogenic heat sources and adiabatic heating exert a major control on both the vigour of SSC and the thermal structure of the lithosphere. We find that either dislocation creep, diffusion creep, or a combination of these mechanisms, can generate SSC with rheological parameters given by laboratory experiments. However, vigorous SSC and significant lithospheric erosion are only possible for relatively low activation energies.

  Abstract
  Sublithospheric small-scale convection (SSC) is thought to be responsible for the flattening of the seafloor depth and surface heat flow observed in mature plates. Although [...]

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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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