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The purpose of this paper is to provide a framework for the implementation of an adjoint sensitivity formulation for least‐squares partial differential equations constrained optimization problems exploiting a multiphysics finite elements package. The estimation of the diffusion coefficient in a Poisson‐type diffusion equation is used as an example.

The authors derive the adjoint formulation in a continuous setting allowing to attribute to the direct and adjoint states the role of different fields to be solved for. They are one‐way coupled through the mismatch between measured and direct states acting as a source term in the adjoint equation. Having solved for the direct and adjoint state, the sensitivity of the cost function with respect to the design variables can then be obtained by a suitable post‐processing procedure. This sensitivity can then be used to efficiently solve the least‐squares problem.

The authors derived the adjoint formulation in a continuous setting allowing the direct and adjoint states to be attributed the role of different fields to be solved. They are one‐way coupled through the mismatch between measured and direct states acting as a source term in the adjoint equation. It is found that, having solved for the direct and adjoint state, the sensitivity of the cost function with respect to the design variables can then be obtained by a suitable post‐processing procedure.

This paper implies that modern multiphysics finite elements packages provide a flexible and extendable software environment for the experimentation with different adjoint formulations. Such tools are therefore expected to become increasingly important in solving notoriously difficult partial differential equation (PDE)‐constrained least‐squares problems. The framework also provides the possibility of experimentation with different regularization techniques (total variation and multiscale techniques for instance) to handle the ill‐posedness of the problem.

In this paper the adjoint sensitivity computation is casted as a multiphysics problem allowing for a flexible and extendable implementation.

The purpose of this paper is to study the influence of the induction heating phenomenon during magnetic pulse forming (MPF) of thin walled tube components. The approach is based on the advanced use of the multiphysics finite element software FORGE® coupling electromagnetism, heat transfer and solid mechanics. Although the global contribution of thermal effects is found to be almost negligible with respect to the volume forces, it can be observed that localized softening due to the heating process induces shock absorbing behavior.

Due to the strong multiphysics couplings between solid mechanics, electromagnetism and heat transfer, it is not always obvious to quantify the contributions of the various physical phenomena. It is thus intended here to take advantage of the numerical framework and tool that has developed to dissociate and quantify the influence of Joule heating phenomena due to eddy currents during MPF processes.

In this paper, the sensitivity of the MPF process has been analyzed to the induction heating source term for a specific tube forming case. An analysis of the electric output signal shows that inductance sensitivity to heating remains small when compared to the mechanical deformation. Regarding mechanical analysis of the process, induction heating contribution has a very slight impact at the global scale, but its effect is more noticeable at the small scale where it is likely that the localized heating induces shock absorption properties through softening. The extension of these results to other materials (for which the thermal dependency of mechanical behavior is different), as well as to a larger range of energy inputs, still needs to be carried out. Such phenomena should be considered for instance for high precision forming.

The analysis of the influence of heating due to eddy currents in magnetic pulse forming processes has not been extensively studied. The originality of this work is to try to quantify its effect on the process by using a numerical-based approach.

The purpose of this paper is to propose a new method that allows to compare the magnetic pressures of different pulse width modulation (PWM) strategies in a fast and efficient way.

The voltage harmonics are determined using the double Fourier integral. As for current harmonics and waveforms, a new generic model based on the Park transformation and a dq model of the machine was established taking saturation into consideration. The obtained analytical waveforms are then injected into a finite element software to compute magnetic pressures using nodal forces.

The overall proposed method allows to accelerate the calculations and the comparison of different PWM strategies and operating points as an analytical model is used to generate current waveforms.

While the analytical expressions of voltage harmonics are already provided in the literature for the space vector pulse width modulation, they had to be calculated for the discontinuous pulse width modulation. In this paper, the obtained expressions are provided. For current harmonics, different models based on a linear and a nonlinear model of the machine are presented in the referenced papers; however, these models are not generic and are limited to the second range of harmonics (two times the switching frequency). A new generic model is then established and used in this paper after being validated experimentally. And finally, the direct injection of analytical current waveforms in a finite element software to perform any magnetic computation is very efficient.

The paper presents the principal elements of automatic adaptivity built in our 2D software for monolithic solution of multiphysics problems based on a fully adaptive finite element method of higher order of accuracy. The adaptive techniques are illustrated by appropriate examples.

Presented are algorithms for realization of the h‐adaptivity, p‐adaptivity, hp‐adaptivity, creation of curvilinear elements for modelling general boundaries and interfaces. Indicated also is the possibility of combining triangular and quadrilateral elements (both classical and curved).

The presented higher‐order adaptive processes are reliable, robust and lead to a substantial reduction of the degrees of freedom in comparison with the techniques used in low‐order finite element methods. They allow solving examples that are by classical approaches either unsolvable or solvable at a cost of high memory and time of computation.

The adaptive processes described in the paper are still limited to 2D computations. Their computer implementation is highly nontrivial (every physical field in a multiphysics task is generally solved on a different mesh satisfying its specific features) and in 3D the number of possible adaptive steps is many times higher.

The described adaptive techniques may represent a powerful tool for the monolithic solution of complex multiphysics problems.

The presented higher‐order adaptive approach of solution is shown to provide better results than the schemes implemented in professional codes based on low‐order finite element methods. Obtaining the results, moreover, requires less time and computer memory.

The purpose of this paper is to present the development and application of a numerical formulation for the structural dynamics and aeroelastic analysis of new generation helicopter and tiltrotor rotor blades. These are characterized by a curvilinear elastic axis, typically with the presence of tip sweep and anhedral angles.

The structural dynamics model implemented is based on nonlinear, flap‐lag‐torsion, rotating beam equations that are valid for slender, homogeneous, isotropic, non‐uniform, twisted blades undergoing moderate displacements. A second‐order approximation scheme for strain‐displacement is adopted. Aerodynamic contributions for aeroelastic applications are derived from sectional theories, with inclusion of wake inflow models to take into account three‐dimensional effects. The numerical integration is obtained through implementation within the COMSOL Multiphysics Finite‐Element‐Method (FEM) software code, considering the elastic axis of arbitrary curvilinear shape.

The computational tool developed is validated by comparisons with results available in the literature. These demonstrate the capability of the tool to accurately predict structural dynamics and aeroelastic behavior of curved‐axis rotor blades. In particular, the influence of sweep and anhedral angles at the blade tip is successfully captured.

The numerical tool developed is limited to the analysis of isotropic blades, with a simple sectional aerodynamic modeling for aeroelastic applications. However, the flexibility of the process through which the proposed tool has been developed is such that a moderate effort is required for its extension to composite blades and more accurate aerodynamic loads predictions.

The proposed computational solver is a reliable tool for preliminary design and optimal design processes of helicopter and tiltrotor rotor blades.

Computational tools for rotors with advanced‐geometry blades are not commonly available. Therefore, the presentation of a successful way to implement structural dynamics/aeroelastic mathematical formulations for rotor blades with curvilinear elastic axis in highly flexible, multiphysics, FEM‐based, commercial software may be of interest for designers and researchers.

This paper gives a bibliographical review of the finite element and boundary element parallel processing techniques from the theoretical and application points of view. Topics include: theory – domain decomposition/partitioning, load balancing, parallel solvers/algorithms, parallel mesh generation, adaptive methods, and visualization/graphics; applications – structural mechanics problems, dynamic problems, material/geometrical non‐linear problems, contact problems, fracture mechanics, field problems, coupled problems, sensitivity and optimization, and other problems; hardware and software environments – hardware environments, programming techniques, and software development and presentations. The bibliography at the end of this paper contains 850 references to papers, conference proceedings and theses/dissertations dealing with presented subjects that were published between 1996 and 2002.

The purpose of this paper is to consider double‐diffusive convection in a heated porous medium saturated with a fluid. Of particular interest is the case where the fluid has a stabilizing concentration gradient and small diffusivity.

A fully‐coupled stabilized finite element scheme and adaptive mesh refinement (AMR) methodology are introduced to solve the resulting coupled multiphysics application and resolve fine scale solution features. The code is written on top of the open source finite element library LibMesh, and is suitable for parallel, high‐performance simulations of large‐scale problems.

The stabilized adaptive finite element scheme is used to compute steady and unsteady onset of convection in a generalized Horton‐Rogers‐Lapwood problem in both two and three‐dimensional domains. A detailed study confirming the applicability of AMR in obtaining the predicted dependence of solutal Nusselt number on Lewis number is given. A semi‐permeable barrier version of the generalized HRL problem is also studied and is believed to present an interesting benchmark for AMR codes owing to the different boundary and internal layers present in the problem. Finally, some representative adaptive results in a complex 3D heated‐pipe geometry are presented.

This work demonstrates the feasibility of stabilized, adaptive finite element schemes for computing simple double‐diffusive flow models, and it represents an easily‐generalizable starting point for more complex calculations since it is based on a highly‐general finite element library. The complementary nature of h‐adaptivity and stabilized finite element techniques for this class of problem is demonstrated using particularly simple error indicators and stabilization parameters. Finally, an interesting double‐diffusive convection benchmark problem having a semi‐permeable barrier is suggested.

Based on the framework of Krylov subspace-based model order reduction (MOR), compact models of the piezoelectric energy harvester devices can be generated. However, the stability of reduced piezoelectric model often cannot be preserved. In previous research studies, “MOR after Schur,” “Schur after MOR” and “multiphysics structure preserving MOR” methods have proven successful in obtaining stable reduced piezoelectric energy harvester models. Though the stability preservation of “MOR after Schur” and “Schur after MOR” methods has already been mathematically proven, the “multiphysics structure preserving MOR” method was not. This paper aims to provide the missing mathematical proof of “multiphysics structure preserving MOR.”

Piezoelectric energy harvesters can be represented by system of differential-algebraic equations obtained by the finite element method. According to the block structure of its system matrices, “MOR after Schur” and “Schur after MOR” both perform Schur complement transformations either before or after the MOR process. For the “multiphysics structure preserving MOR” method, the original block structure of the system matrices is preserved during MOR. 

This contribution shows that, in comparison to “MOR after Schur” and “Schur after MOR” methods, “multiphysics structure preserving MOR” method performs the Schur complement transformation implicitly, and therefore, stabilizes the reduced piezoelectric model.

The stability preservation of the reduced piezoelectric energy harvester model obtained through “multiphysics structure preserving MOR” method is proven mathematically and further validated by numerical experiments on two different piezoelectric energy harvester devices.

This aim of this work is to investigate different modelling approaches for air-cooled data centres. The study employs three computational methods, which are based on finite element, finite volume and lattice Boltzmann methods and which are respectively implemented via commercial Multiphysics software, open-source computational fluid dynamics code and graphical processing unit-based code developed by the authors. The results focus on comparison of the three methods, all of which include models for turbulence, when applied to two rows of datacom racks with cool air supplied via an underfloor plenum.

This paper studies thermal airflows in a data centre by applying different numerical simulation techniques that are able to analyse the thermal airflow distribution for a simplified layout of datacom racks in the presence of a computer room air conditioner.

Good quantitative agreement between the three methods is seen in terms of the inlet temperatures to the datacom equipment. The computational methods are contrasted in terms of application to thermal management of data centres.

The work demonstrates how the different simulation techniques applied to thermal management of airflow in a data centre can provide valuable design and operational understanding. Basing the analysis on three very different computational approaches is new and would offer an informed understanding of their potential for a class of problems.

The purpose of this paper is to numerically investigate the effects of the shape on the performance of the cathode catalyst agglomerate used in polymer electrolyte fuel cells (PEFCs). The shapes investigated are slabs, cylinders and spheres.

Three 1D models are developed to represent the slab like, cylindrical and spherical agglomerates, respectively. The models are solved for the concentration of the dissolved oxygen using a finite element software, COMSOL Multiphysics®. “1D” and “1D axisymmetric” schemes are used to model the slab like and cylindrical agglomerates, respectively. There is no one-dimensional scheme available in COMSOL Multiphysics® for spherical coordinate systems. To resolve this, the governing equation in “1D” scheme is mathematically modified to match that of the spherical coordinate system.

For a given length of the diffusion path, the variation in the performances of the investigated agglomerates is dependent on the operational overpotential. Under low magnitudes of the overpotentials, where the performance is mainly limited by reaction, the slab-like agglomerate outperforms the spherical and cylindrical agglomerates. In contrast, under high magnitudes of the overpotentials where the agglomerate performance is mainly limited by diffusion, the spherical and cylindrical agglomerates outperform the slab-like agglomerate.

The current advances in the nano-fabrication technology gives more flexibility in designing the catalyst layers in PEFCs to the desired structures. If the design of the agglomerate catalyst is to be assessed, the current micro-scale modelling offers an efficient and rapid way forward.

The current micro-scale modelling is an efficient alternative to developing a full (or half) fuel cell model to evaluate the effects of the agglomerate structure.