69 research outputs found
An X-FEM and Level Set computational approach for image-based modeling. Application to homogenization.
International audienceThe advances in material characterization by means of imaging techniques require powerful computational methods for numerical analysis. The present contribution focuses on highlighting the advantages of coupling the Extended Finite Elements Method (X-FEM) and the level sets method, applied to solve microstructures with complex geometries. The process of obtaining the level set data starting from a digital image of a material structure and its input into an extended finite element framework is presented. The coupled method is validated using reference examples and applied to obtain homogenized properties for heterogeneous structures. Although the computational applications presented here are mainly two dimensional, the method is equally applicable for three dimensional problems
Accuracy analysis of structure with nearby interfaces within XFEM
© 2017 Author(s). This paper presents the fundamental principle of the extended finite element method (XFEM) for electromagnetic field analysis. The accuracy analysis of structure with nearby interfaces within XFEM is presented. A numerical example applied to the parallel plate electrodes in 1-D static electric field is provided. Two types of meshing are used to analyse the accuracy of the meshing where the support of the enriched node are cut by more than one interface
Three embedded techniques for finite element heat flow problem with embedded discontinuities
The final publication is available at Springer via http://dx.doi.org/10.1007/s00466-017-1382-7The present paper explores the solution of a heat conduction problem considering discontinuities embedded within the mesh and aligned at arbitrary angles with respect to the mesh edges. Three alternative approaches are proposed as solutions to the problem. The difference between these approaches compared to alternatives, such as the eXtended Finite Element Method (X-FEM), is that the current proposal attempts to preserve the global matrix graph in order to improve performance. The first two alternatives comprise an enrichment of the Finite Element (FE) space obtained through the addition of some new local degrees of freedom to allow capturing discontinuities within the element. The new degrees of freedom are statically condensed prior to assembly, so that the graph of the final system is not changed. The third approach is based on the use of modified FE-shape functions that substitute the standard ones on the cut elements. The imposition of both Neumann and Dirichlet boundary conditions is considered at the embedded interface. The results of all the proposed methods are then compared with a reference solution obtained using the standard FE on a mesh containing the actual discontinuity.Peer ReviewedPostprint (author's final draft
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Advances in Design and Optimization Using Immersed Boundary Methods
This thesis is concerned with topology optimization which provides engineers with a systematic approach to optimize the layout and geometry of a structure against various design criteria. Traditional topology optimization uses density-based methods to capture topological changes in geometry. Density-based methods describe a structural layout using artificial elemental densities. To obtain a good resolution of the geometry, fine meshes are required. This however leads to large computational costs in 3D. Using coarser but practical meshes results in blurred structural boundaries and unreliable prediction of physical response along those boundaries. Using immersed boundary methods instead, such as the extended finite element method (XFEM), alleviates these issues. The XFEM provides clear description of the geometry, and approximation of the physical response along boundaries has been shown to converge to the approximation using body-fitted meshes. This thesis focuses on the use of XFEM for topology optimization. Design geometry in this thesis is tracked precisely using the level set method (LSM).The LSM-XFEM approach is used to solve variety of multiphysics design and optimization problems. However, being a relatively new field of study the LSM-XFEM approach continues to pose many interesting challenges limiting its applicability to topology optimization. The goal of this thesis is to present advances made towards making LSM-XFEM more viable and reliable for design and optimization of multiphysics problems. Specifically, i) The numerical behavior of XFEM-based shape sensitivities has not yet been investigated. This thesis presents a first-of-its kind study on the numerical behavior of shape sensitivities using the XFEM. ii) The matter of overestimation of stresses using the XFEM, a longstanding issue with no concrete resolution available in the literature, is addressed for robust stress-based optimization. iii) LSM-based topology optimization is known to suffer from slow design evolution resulting from localized sensitivities. A recently proposed concept of geometric primitives as design variables alleviates this issue. Literature on this concept has been restricted to single material problems using linear elasticity. Using the XFEM, this thesis extends the concept of geometric primitives as design variables to multiphase multiphysics problems in 3D
Inverse und Optimierungsprobleme für piezoelektrische Materialien mit der Extended Finite Elemente Methode und Level sets
Piezoelectric materials are used in several applications as sensors and actuators where they experience high stress and electric field concentrations as a result of which they may fail due to fracture. Though there are many analytical and experimental works on piezoelectric fracture mechanics. There are very few studies about damage detection, which is an interesting way to prevent the failure of these ceramics.
An iterative method to treat the inverse problem of detecting cracks and voids in piezoelectric structures is proposed. Extended finite element method (XFEM) is employed for solving the inverse problem as it allows the use of a single regular mesh for large number of iterations with different flaw geometries.
Firstly, minimization of cost function is performed by Multilevel Coordinate Search (MCS) method. The XFEM-MCS methodology is applied to two dimensional electromechanical problems where flaws considered are straight cracks and elliptical voids. Then a numerical method based on combination of classical shape derivative and level set method for front propagation used in structural optimization is utilized to minimize the cost function. The results obtained show that the XFEM-level set methodology is effectively able to determine the number of voids in a piezoelectric structure and its corresponding locations.
The XFEM-level set methodology is improved to solve the inverse problem of detecting inclusion interfaces in a piezoelectric structure. The material interfaces are implicitly represented by level sets which are identified by applying regularisation using total variation penalty terms. The formulation is presented for three dimensional structures and inclusions made of different materials are detected by using multiple level sets. The results obtained prove that the iterative procedure proposed can determine the location and approximate shape of material subdomains in the presence of higher noise levels.
Piezoelectric nanostructures exhibit size dependent properties because of surface elasticity and surface piezoelectricity. Initially a study to understand the influence of surface elasticity on optimization of nano elastic beams is performed. The boundary of the nano structure is implicitly represented by a level set function, which is considered as the design variable in the optimization process. Two objective functions, minimizing the total potential energy of a nanostructure subjected to a material volume constraint and minimizing the least square error compared to a target
displacement, are chosen for the numerical examples. The numerical examples demonstrate the importance of size and aspect ratio in determining how surface effects impact the optimized topology of nanobeams.
Finally a conventional cantilever energy harvester with a piezoelectric nano layer is analysed. The presence of surface piezoelectricity in nano beams and nano plates leads to increase in electromechanical coupling coefficient. Topology optimization of these piezoelectric structures in an energy harvesting device to further increase energy conversion using appropriately modified XFEM-level set algorithm is performed
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Topology Optimization of Flow Problems Modeled by the Incompressible Navier-Stokes Equations
This work is concerned with topology optimization of incompressible flow problems. While size and shape optimization methods are limited to modifying existing boundaries, topology optimization allows for merging boundaries as well as creating new ones. Since topology optimization methods do not require a good initial guess, they are powerful tools for finding new and non-intuitive designs. The latter is particularly beneficial for flow problems which are typically nonlinear as well as transient. Depending on the complexity of the flow problem, predicting a solution may be challenging. Determining an improved or optimized design for complex flow problems is an even greater challenge as it not only requires a solution to the flow problem for a given design, but also a prediction on how a design change will affect the flow. Fluid topology optimization commonly uses a material interpolation approach for describing the geometry during the optimization process: solid material is modeled via an artificial porosity that penalizes the flow velocities. While this approach works well for simple steady-state problems aiming to minimize the dissipated energy, the current study shows that using the porosity approach may cause issues for more complex problems such as coupled fluid-structure-interaction (FSI) systems, unsteady flow problems or problems aiming to match a target performance. To overcome these issues a geometric boundary description based on level sets is developed. This geometric boundary description is applied to both, a steady-state hydrodynamic lattice Boltzmann formulation and a stabilized finite element formulation of the steady-state Navier-Stokes equations. The enforcement of the no-slip condition along the fluid-solid interface is handled via an immersed boundary technique in case of the lattice Boltzmann method, while the Navier-Stokes formulation uses an extended finite element method (XFEM). Through the research conducted in this work, the spectrum of flow problems that can be solved by topology optimization techniques has been broadened significantly
Analytical sensitivity analysis using the extended finite element method in shape optimization of bimaterial structures
peer reviewedThe present work investigates the shape optimization of bimaterial structures. The problem is formulated using a level set description of the geometry and the extended finite element method (XFEM) to enable an easy treatment of complex geometries. A key issue comes from the sensitivity analysis of the structural responses with respect to the design parameters ruling the boundaries. Even if the approach does not imply any mesh modification, the study shows that shape modifications lead to difficulties when the perturbation of the level sets modifies the set of extended finite elements. To circumvent the problem, an analytical sensitivity analysis of the structural system is developed. Differences between the sensitivity analysis using FEM or XFEM are put in evidence. To conduct the sensitivity analysis, an efficient approach to evaluate the so-called velocity field is developed within the XFEM domain. The proposed approach determines a continuous velocity field in a boundary layer around the zero level set using a local finite element approximation. The analytical sensitivity analysis is validated against the finite differences and a semi- analytical approach. Finally our shape optimization tool for bimaterial structures is illustrated by revisiting the classical problem of the shape of soft and stiff inclusions in plates
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