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Computational mechanics Finite element method Image processing Parallel computing Polynomial chaos expansion Stabilized finite element methods a posteriori error estimation,reduced basis Goal-oriented error assessment Orthogonal sub-grid scales 3D tumor platforms 41A25 65N30 65N38 65N50 65R20#### Author

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- Chinesta, F. 10 (%)
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## BEM-FVM solution of the conjugate radiative and convective heat transfer problem

### Archives of Computational Methods in Engineering (2006-06-01) 13: 171-248 , June 01, 2006

### Summary

The problem addressed in the paper is the coupling between heat radiation and convection in participating media. While convection is modelled by finite volumes, heat radiation is solved using the boundary element method (BEM). The latter is a technique of solving the integral equations of radiation using weighted residuals.

BEM can be seen as an alternative approach to the well established zoning method or FEM, its higher order generalization. When compared with these approaches, BEM offers substantial computing time economy due to the reduction of the integration dimension and lack of volume integrals.

Coupling convection solution with heat radiation is accomplished in an iterative way. First the initial temperatures of the medium is computed by the convection solver for given walls temperature assuming no interaction with radiation. Using this temperature field the radiative heat fluxes and sources are computed and their values substituted to the corrected energy convection equation. The procedure is repeated until a required accuracy is reached. Mild underrelaxaction of the heat sources improves the convergence.

BEM radiation procedure requires numerical integration over all discrete surface elements, and ray tracing of the Gaussian rays connecting the collocation point and the nodes of the Gaussian quadrature. The latter is the most time consuming operation. Numerical tests have shown that standard ray tracing on convection meshes leads to prohibitively long computing time. To accelerate the procedure the ray tracing is performed on a much coarser structural grid. This is an acceptable approximation as heat radiation volumetric grid does not need to capture small scale phenomena which is in contrast with the convection grid where the resolution of the resulting fields depends strongly on the mesh density. This assumption accelerates the ray tracing by at least two orders of magnitude. The transition between the radiative and convection nodes is accomplished using the radial basis function network concept.

Several industrial problems have been solved using this model. Commercial CFD code Fluent has been used to solve the convection equations. The interaction between the in-house radiative code BERTA and Fluent was maintained by modifying source term of energy balance equation within latter. The coupling was programmed at a level of a script.

The results have been compared with some available benchmark solutions and with the radiative transfer solvers (Discrete Ordinates and Discrete Transfer) installed in the CFD code. Very good agreement has been observed.

The ray tracing concept has been extended to cylindrical coordinates systems to solve axisymmetric problems. The technique has been also used to model the interaction of radiation and conduction in semitransparent, non gray media.

Numerical results of both some benchmark solutions and industrial problems are shown in the paper.

## Challenges in Thermo-mechanical Analysis of Friction Stir Welding Processes

### Archives of Computational Methods in Engineering (2017-01-01) 24: 189-225 , January 01, 2017

This paper deals with the numerical simulation of friction stir welding (FSW) processes. FSW techniques are used in many industrial applications and particularly in the aeronautic and aerospace industries, where the quality of the joining is of essential importance. The analysis is focused either at global level, considering the full component to be jointed, or locally, studying more in detail the heat affected zone (HAZ). The analysis at global (structural component) level is performed defining the problem in the Lagrangian setting while, at local level, an apropos kinematic framework which makes use of an efficient combination of Lagrangian (pin), Eulerian (metal sheet) and ALE (stirring zone) descriptions for the different computational sub-domains is introduced for the numerical modeling. As a result, the analysis can deal with complex (non-cylindrical) pin-shapes and the extremely large deformation of the material at the HAZ without requiring any remeshing or remapping tools. A fully coupled thermo-mechanical framework is proposed for the computational modeling of the FSW processes proposed both at local and global level. A staggered algorithm based on an isothermal fractional step method is introduced. To account for the isochoric behavior of the material when the temperature range is close to the melting point or due to the predominant deviatoric deformations induced by the visco-plastic response, a mixed finite element technology is introduced. The Variational Multi Scale method is used to circumvent the LBB stability condition allowing the use of linear/linear P1/P1 interpolations for displacement (or velocity, ALE/Eulerian formulation) and pressure fields, respectively. The same stabilization strategy is adopted to tackle the instabilities of the temperature field, inherent characteristic of convective dominated problems (thermal analysis in ALE/Eulerian kinematic framework). At global level, the material behavior is characterized by a thermo–elasto–viscoplastic constitutive model. The analysis at local level is characterized by a rigid thermo–visco-plastic constitutive model. Different thermally coupled (non-Newtonian) fluid-like models as Norton–Hoff, Carreau or Sheppard–Wright, among others are tested. To better understand the material flow pattern in the stirring zone, a (Lagrangian based) particle tracing is carried out while post-processing FSW results. A coupling strategy between the analysis of the process zone nearby the pin-tool (local level analysis) and the simulation carried out for the entire structure to be welded (global level analysis) is implemented to accurately predict the temperature histories and, thereby, the residual stresses in FSW.

## High Order Direct Arbitrary-Lagrangian–Eulerian (ALE) Finite Volume Schemes for Hyperbolic Systems on Unstructured Meshes

### Archives of Computational Methods in Engineering (2017-11-01) 24: 751-801 , November 01, 2017

In this work we develop a new class of high order accurate Arbitrary-Lagrangian–Eulerian (ALE) one-step finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering *moving* unstructured triangular and tetrahedral meshes, respectively. As usual for finite volume schemes, data are represented within each control volume by piecewise constant values that evolve in time, hence implying the use of some strategies to improve the order of accuracy of the algorithm. In our approach high order of accuracy in space is obtained by adopting a *WENO reconstruction* technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space–time *finite element predictor*, which performs a one-step time discretization. Specifically, we adopt a discontinuous Galerkin predictor which can handle stiff source terms that might produce jumps in the local space–time solution. Since we are dealing with moving meshes the elements deform while the solution is evolving in time, hence making the use of a reference system very convenient. Therefore, within the space–time predictor, the physical element is mapped onto a reference element using a high order isoparametric approach, where the space–time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space–time nodes. The computational mesh continuously changes its configuration in time, following as closely as possible the flow motion. The entire mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. In order to obtain a continuous mesh configuration at any time level, the mesh motion is evaluated by assigning each node of the computational mesh with a unique velocity vector at each timestep. The *nodal solver* algorithm preforms the Lagrangian stage, while we rely on a *rezoning algorithm* to improve the mesh quality when the flow motion becomes very complex, hence producing highly deformed computational elements. A so-called *relaxation algorithm* is finally employed to partially recover the optimal Lagrangian accuracy where the computational elements are not distorted too much. We underline that our scheme is supposed to be an ALE algorithm, where the local mesh velocity can be chosen independently from the local fluid velocity. Once the vertex velocity and thus the new node location has been determined, the old element configuration at time
$$t^n$$
is connected with the new one at time
$$t^{n+1}$$
with *straight edges* to represent the local mesh motion, in order to maintain algorithmic simplicity. The final *ALE finite volume scheme* is based directly on a space–time conservation formulation of the governing system of hyperbolic balance laws. The nonlinear system is reformulated more compactly using a space–time divergence operator and is then integrated on a moving space–time control volume. We adopt a linear parametrization of the space–time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. We apply the new high order direct ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical magneto-hydrodynamics equations and the non-conservative seven-equation Baer–Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes. Finally we briefly present some variants of the algorithm that aim at improving the overall computational efficiency.

## Modelling Techniques for Vibro-Acoustic Dynamics of Poroelastic Materials

### Archives of Computational Methods in Engineering (2015-04-01) 22: 183-236 , April 01, 2015

Given the quest for mass reduction while preserving proper vibration and acoustic comfort levels in industrial machinery and vehicles, lightweight poroelastic materials have gained a lot of importance. Often, these materials are applied in a multilayered configuration, which can consist of a number of acoustic, elastic, viscoelastic and poroelastic layers. Among these, poroelastic materials are the main focus of this paper. A poroelastic material comprises two constituents, being the elastic solid constituent, also called the frame, and the fluid filling the voids. Depending on the frequency range of interest, the motion of both constituents can be strongly coupled. Poroelastic materials can dissipate energy very effectively by structural, thermal and viscous means. Considerable research effort has been put in the development of robust models and prediction techniques which are capable of accurately describing the damping phenomena of these materials. After a broad introduction, this paper reviews the most commonly used models, ranging from simple empirical relations to detailed models accounting for the coupled behaviour of both phases and the CAE modelling techniques currently being applied for the analysis of the time-harmonic vibro-acoustic behaviour of these materials. Commonly used methods, such as the Finite Element Method and the Transfer Matrix Method which are mainly fitted for low-freqency and high-frequency applications, respectively, are discussed as well as extensions to improve their efficiency and applicability. The two final sections pay special attention to the promising Wave Based Method, a Trefftz-based technique, the application range of which was recently extended towards poroelastic problems.

## A Review on the Mechanical Modeling of Composite Manufacturing Processes

### Archives of Computational Methods in Engineering (2017-04-01) 24: 365-395 , April 01, 2017

The increased usage of fiber reinforced polymer composites in load bearing applications requires a detailed understanding of the process induced residual stresses and their effect on the shape distortions. This is utmost necessary in order to have more reliable composite manufacturing since the residual stresses alter the internal stress level of the composite part during the service life and the residual shape distortions may lead to not meeting the desired geometrical tolerances. The occurrence of residual stresses during the manufacturing process inherently contains diverse interactions between the involved physical phenomena mainly related to material flow, heat transfer and polymerization or crystallization. Development of numerical process models is required for virtual design and optimization of the composite manufacturing process which avoids the expensive trial-and-error based approaches. The process models as well as applications focusing on the prediction of residual stresses and shape distortions taking place in composite manufacturing are discussed in this study. The applications on both thermoset and thermoplastic based composites are reviewed in detail.

## Recent Developments in Variational Multiscale Methods for Large-Eddy Simulation of Turbulent Flow

### Archives of Computational Methods in Engineering (2017-02-27): 1-44 , February 27, 2017

The variational multiscale method is reviewed as a framework for developing computational methods for large-eddy simulation of turbulent flow. In contrast to other articles reviewing this topic, which focused on large-eddy simulation of turbulent incompressible flow, this study covers further aspects of numerically simulating turbulent flow as well as applications beyond incompressible single-phase flow. The various concepts for subgrid-scale modeling within the variational multiscale method for large-eddy simulation proposed by researchers in this field to date are illustrated. These conceptions comprise (i) implicit large-eddy simulation, represented by residual-based and stabilized methods, (ii) functional subgrid-scale modeling via small-scale subgrid-viscosity models and (iii) structural subgrid-scale modeling via the introduction of multifractal subgrid scales. An overview on exemplary numerical test cases to which the reviewed methods have been applied in the past years is provided, including explicit computational results obtained from turbulent channel flow. Wall-layer modeling, passive and active scalar transport as well as developments for large-eddy simulation of turbulent two-phase flow and combustion are discussed to complete this exposition.

## Reduced Basis’ Acquisition by a Learning Process for Rapid On-line Approximation of Solution to PDE’s: Laminar Flow Past a Backstep

### Archives of Computational Methods in Engineering (2017-08-05): 1-11 , August 05, 2017

Reduced basis methods for the approximation to parameter-dependent partial differential equations are now well-developed and start to be used for industrial applications. The classical implementation of the reduced basis method goes through two stages: in the first one, offline and time consuming, from standard approximation methods a reduced basis is constructed; then in a second stage, online and very cheap, a small problem, of the size of the reduced basis, is solved. The offline stage is a learning one from which the online stage can proceed efficiently. In this paper we propose to exploit machine learning procedures in both offline and online stages to either tackle different classes of problems or increase the speed-up during the online stage. The method is presented through a simple flow problem—a flow past a backward step governed by the Navier Stokes equations—which shows, however, interesting features.

## Nonlinear Shape-Manifold Learning Approach: Concepts, Tools and Applications

### Archives of Computational Methods in Engineering (2016-09-08): 1-21 , September 08, 2016

In this paper, we present the concept of a “shape manifold” designed for reduced order representation of complex “shapes” encountered in mechanical problems, such as design optimization, springback or image correlation. The overall idea is to define the shape space within which evolves the boundary of the structure. The reduced representation is obtained by means of determining the intrinsic dimensionality of the problem, independently of the original design parameters, and by approximating a hyper surface, i.e. a shape manifold, connecting all admissible shapes represented using level set functions. Also, an optimal parameterization may be obtained for arbitrary shapes, where the parameters have to be defined a posteriori. We also developed the predictor-corrector optimization *manifold walking* algorithms in a reduced shape space that guarantee the admissibility of the solution with no additional constraints. We illustrate the approach on three diverse examples drawn from the field of computational and applied mechanics.

## Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems

### Archives of Computational Methods in Engineering (2010-12-01) 17: 403-434 , December 01, 2010

Uncertainty quantification and propagation in physical systems appear as a critical path for the improvement of the prediction of their response. Galerkin-type spectral stochastic methods provide a general framework for the numerical simulation of physical models driven by stochastic partial differential equations. The response is searched in a tensor product space, which is the product of deterministic and stochastic approximation spaces. The computation of the approximate solution requires the solution of a very high dimensional problem, whose calculation costs are generally prohibitive. Recently, a model reduction technique, named Generalized Spectral Decomposition method, has been proposed in order to reduce these costs. This method belongs to the family of Proper Generalized Decomposition methods. It takes part of the tensor product structure of the solution function space and allows the *a priori* construction of a quasi optimal separated representation of the solution, which has quite the same convergence properties as *a posteriori* Hilbert Karhunen-Loève decompositions. The associated algorithms only require the solution of a few deterministic problems and a few stochastic problems on deterministic reduced basis (algebraic stochastic equations), these problems being uncoupled. However, this method does not circumvent the “curse of dimensionality” which is associated with the dramatic increase in the dimension of stochastic approximation spaces, when dealing with high stochastic dimension. In this paper, we propose a marriage between the Generalized Spectral Decomposition algorithms and a separated representation methodology, which exploits the tensor product structure of stochastic functions spaces. An efficient algorithm is proposed for the *a priori* construction of separated representations of square integrable vector-valued functions defined on a high-dimensional probability space, which are the solutions of systems of stochastic algebraic equations.

## Meshless Finite Difference Method with Higher Order Approximation—Applications in Mechanics

### Archives of Computational Methods in Engineering (2012-03-01) 19: 1-49 , March 01, 2012

This work is devoted to some recent developments in the Higher Order Approximation introduced to the Meshless Finite Difference Method (MFDM), and its application to the solution of boundary value problems in mechanics. In the MFDM, approximation of the sought function is described in terms of nodes rather than by means of any imposed structure like elements, regular meshes etc. Therefore, the MFDM, using arbitrarily irregular clouds of nodes using the Moving Weighted Least Squares (MWLS) approximation falls into the category of the Meshless Methods (MM). The MFDM, dating to early seventies, is one of the oldest and possibly the most developed one. In this paper considered are some techniques which lead to improvement of the MFDM solution’s quality. The main objective of this paper is the presentation and overview of new ideas and the development of the Higher Order solution approach in the MFDM provided by correction terms, preceded by a brief information about the current state-of-the art of this method. The main concept of the Higher Order Approximation (HOA) used here, is based on consideration of additional terms in the local Taylor expansion of the sought function. It shall be demonstrated that such a move may essentially improve, in many ways, efficiency and solution quality of the Higher Order MFDM. The Higher Order correction terms may be applied in many aspects of the MFDM solution approach. Among them one may distinguish the a-posteriori error estimation as well as adaptive solution process with multigrid strategy. Moreover, in the present work considered are: computational implementation of the Higher Order MFDM algorithms, examination of the above mentioned aspects using 1D and 2D benchmark tests, as well as an application of the Higher Order MFDM solution approach to selected boundary value problems in mechanics.