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See detailA Bayesian inversion approach to recovering material parameters in hyperelastic solids using dolfin-adjoint
Hale, Jack UL; Farrell, Patrick E.; Bordas, Stéphane UL

Presentation (2015, July 01)

In the first part of the talk I will describe in general terms the link between classical optimisation techniques and the Bayesian approach to statistical inversion as outlined in the seminal book of ... [more ▼]

In the first part of the talk I will describe in general terms the link between classical optimisation techniques and the Bayesian approach to statistical inversion as outlined in the seminal book of [Kaipio and Somersalo, 2005]. Under the assumption of an additive Gaussian noise model, a Gaussian prior distribution and a linear parameter-to-observable map, it is possible to uniquely characterise the Bayesian posterior as Gaussian with the maximum aposteriori (MAP) point equal to the minimum of a classic regularised minimisation problem and covariance matrix equal to the inverse of the Hessian of the functional evaluated at the MAP point. I will also discuss techniques that can be used when these assumptions break down. In the second part of the talk I will describe a method implemented within dolfin-adjoint [Funke and Farrell, arXiv 2013] to quantify the uncertainty in the recovered material parameters of a hyperelastic solid from partial and noisy observations of the displacement field in the domain. The finite element discretisation of the adjoint and higher-order adjoint (Hessian) equations are derived automatically from the high-level UFL representation of the problem. The resulting equations are solved using PETSc. I will concentrate on finding the eigenvalue decomposition of the posterior covariance matrix (Hessian). The eigenvectors associated with the lowest eigenvalues of the Hessian correspond with the directions in parameter space least constrained by the observations [Flath et al. 2011]. This eigenvalue problem is tricky to solve efficiently because the Hessian is very large (on the order of the number of parameters) and dense (meaning that only its action on a vector can be calculated, each involving considerable expense). Finally, I will show some illustrative examples including the uncertainty associated with deriving the material properties of a 3D hyperelastic block with a stiff inclusion with knowledge only of the displacements on the boundary of the domain. J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, vol. 160. New York: Springer-Verlag, 2005. S. W. Funke and P. E. Farrell, “A framework for automated PDE-constrained optimisation,” arXiv:1302.3894 [cs], Feb. 2013. H. P. Flath, L. C. Wilcox, V. Akçelik, J. Hill, B. van Bloemen Waanders, and O. Ghattas, “Fast Algorithms for Bayesian Uncertainty Quantification in Large-Scale Linear Inverse Problems Based on Low-Rank Partial Hessian Approximations,” SIAM J. Sci. Comput., vol. 33, no. 1, pp. 407–432, Feb. 2011. [less ▲]

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See detailfenics-shells: a UFL-based library for simulating thin structures
Brunetti, Matteo; Hale, Jack UL; Bordas, Stéphane UL et al

Scientific Conference (2015, July 01)

Shell, plate and beam (thin) structures are widely used in civil, mechanical and aeronautical engineering because they are capable of carrying high loads with a minimal amount of structural mass. Because ... [more ▼]

Shell, plate and beam (thin) structures are widely used in civil, mechanical and aeronautical engineering because they are capable of carrying high loads with a minimal amount of structural mass. Because the out-of-plane dimension is usually much smaller than the two in-plane dimensions, it is possible to asymptotically reduce the full 3D equations of elasticity to a whole variety of equivalent 2D models posed on a manifold embedded in 3D space. This reduction results in massively reduced computational expense and remains a necessity for practical large-scale computation of structures of real engineering interest such as the fuselage of an aircraft. The numerical solution of such mathematical models is a challenging task, especially for very thin shells when shear and membrane locking effects require special attention. As originally noted by [Hale and Baiz, 2013], the high-level form language UFL provides an excellent framework for writing extensible, reusable and pedagogical numerical models of thin structures. To our knowledge fenics-shells represents the first unified open-source implementation of a wide range of thin structural models, including Reissner-Mindlin, Kirchhoff-Love, Von Karman and hierarchical (higher-order) plates, and Madare-Naghdi and Madare-Koiter shell models. Because of the broad scope of fenics-shells, in this talk we will focus on how to cure numerical locking by applying the Mixed Interpolation of Tensorial Components (MITC) approach of [Dvorkin and Bathe, 1986] and [Lee and Bathe, 2010] to a shell with an initially flat reference configuration. The MITC approach consists of an element-by-element interpolation of the degrees of freedom of the rotations onto the degrees of freedom of a reduced rotation space, the latter typically constructed using H(curl) conforming finite elements such as the rotated Raviart-Thomas-Nédélec elements. Then, the bilinear form is constructed on the underlying H(curl) space. Because of the interpolation operator, the original problem is expressed in terms of the degrees of freedom for the rotations only. Within DOLFIN we have implemented this projection operation using two UFL forms within a custom assembler compiled just-in-time using Instant. We show numerical convergence studies that match the apriori bounds available in the literature. E. N. Dvorkin and K.-J. Bathe, “A continuum mechanics based four-node shell element for general non-linear analysis,” Engineering Computations, vol. 1, no. 1, pp. 77–88, 1984. P. S. Lee and K. J. Bathe, “The quadratic MITC plate and MITC shell elements in plate bending,” Advances in Engineering Software, vol. 41, no. 5, pp. 712–728, 2010. J. S. Hale and P. M. Baiz, “Towards effective shell modelling with the FEniCS project” presented at the FEniCS Conference 2013, Jesus College, Cambridge, 19-Mar-2013. [less ▲]

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See detailLarge scale phase field model of fracture and cutting in soft tissues
Ziael-Rad, Vahid; Hale, Jack UL; Maurini, Corrado et al

Scientific Conference (2015, July)

The phase field method has proven to be an important tool in computational mechanics in that it is able to deal naturally with crack nucleation and branching [1]. In this contribution, we demonstrate a ... [more ▼]

The phase field method has proven to be an important tool in computational mechanics in that it is able to deal naturally with crack nucleation and branching [1]. In this contribution, we demonstrate a large scale phase field model of fracture and cutting of soft tissues undergoing non-linear deformations with a material law defined by a hyperelastic energy density functional. We will also provide some initial thoughts on the how the effect of a porous medium can be incorporated into the phase field model. We implement this work using the FEniCS project and PETSc software packages [2, 3]. [less ▲]

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See detailExtended Finite Element Method with Global Enrichment
Agathos, Konstantinos; Chatzi, Eleni; Bordas, Stéphane UL et al

Scientific Conference (2015, July)

A variant of the extended finite element method is presented which facilitates the use of enriched elements in a fixed volume around the crack front (geometrical enrichment) in 3D fracture problems. The ... [more ▼]

A variant of the extended finite element method is presented which facilitates the use of enriched elements in a fixed volume around the crack front (geometrical enrichment) in 3D fracture problems. The major problem associated with geometrical enrichment is that it significantly deteriorates the conditioning of the resulting system matrices, thus increasing solution times and in some cases making the systems unsolvable. For 2D problems this can be dealt with by employing degree of freedom gathering [1] which essentially inhibits spatial variation of enrichment function weights. However, for the general 3D problem such an approach is not possible since spatial variation of the enrichment function weights in the direction of the crack front is necessary in order to reproduce the variation of solution variables, such as the stress intensity factors, along the crack front. The proposed method solves the above problem by employing a superimposed mesh of special elements which serve as a means to provide variation of the enrichment function weights along the crack front while still not allowing variation in any other direction. The method is combined with special element partitioning algorithms [2] and numerical integration schemes [3] as well as techniques for the elimination of blending errors between the standard and enriched part of the approximation in order to further improve the accuracy of the produced results. Additionally, a novel benchmark problem is introduced which enables the computation of displacement and energy error norms as well as errors in the stress intensity factors for the general 3D case. Through this benchmark problem it is shown that the proposed method provides optimal convergence rates, improved accuracy and reduced computational cost compared to standard XFEM. [less ▲]

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See detailAdvances in error estimation for homogenisation
Alves Paladim, Daniel; Kerfriden, Pierre; Moitinho de Almeida, José Paulo et al

in 13th U.S. National Congress on Computational Mechanics (2015, July)

In this paper, the concept of modeling error is extended to the homogenisation of elliptic PDEs. The main difficulty is the lack of a full description of the diffusion coefficients. We overcome this ... [more ▼]

In this paper, the concept of modeling error is extended to the homogenisation of elliptic PDEs. The main difficulty is the lack of a full description of the diffusion coefficients. We overcome this obstacle by representing them as a random a field. Under this framework, it is possible to quantify the accuracy of the surrogate model (the homogenised model) in terms of first moments of the energy norm and quantities of interest. This work builds on the seminal work of [1]. The methodology here presented rely on the Constitutive Relation Error (CRE) which states that a certain measures of the primal and dual surrogate model upper bound the exact error. The surrogate model, in agreement with homogenisation, is deterministic. This property exploited to obtain bounds whose computation is also deterministic. It is also shown that minimising the CRE in the set of homogenisation schemes leads us to an optimal surrogate that is closely related to the classical Voigt and Reuss models. Numerical examples demonstrate that the bounds are easy and affordable to compute, and useful as long as the mismatch between he diffusion coefficients of the microstructure remain small. In the case of high mismatch, extensions are proposed, through the introduction of stochastic surrogate models.. [1]Romkes, Albert, J. Tinsley Oden, and Kumar Vemaganti."Multi-scale goal-oriented adaptive modeling of random heterogeneous materials." Mechanics of materials 38.8(2006): 859-872. [less ▲]

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See detailHyperelastic Elastography in a Large-Scale Bayesian Inversion Setting
Hale, Jack UL; Farrel, Patrick E.; Bordas, Stéphane UL

Scientific Conference (2015, July)

We consider the problem of recovering the material parameters of a hyperelastic material [1] in the Bayesian inversion setting. In the Bayesian setting we can extract the statistics associated with ... [more ▼]

We consider the problem of recovering the material parameters of a hyperelastic material [1] in the Bayesian inversion setting. In the Bayesian setting we can extract the statistics associated with various sources of uncertainty, including noise, experimental deficiencies and incomplete observations of the domain. This will allow medical practitioners to make superior diagnosis decisions when presented with a quantitative measure of uncertainty in the recovered parameters. On the assumption of a Gaussian additive noise model, a Gaussian prior and a linear forward model, the posterior distribution of the material parameters given the observations will also be Gaussian. To ensure that the assumption of a linear forward model is valid, and that the posterior is approximated sufficiently well by a Gaussian distribution, we place a limit on the strain regime in which our current methodology applies. We are developing MCMC methods for exploring the non-Gaussian statistics of the posterior distribution. In the linear case, the covariance matrix of the posterior distribution is then characterised by the inverse of the Hessian of the objective functional evaluated at its minimiser. To extract statistical information from the large and dense Hessian we perform a low-rank approximation of the Hessian [2]. The eigenvectors associated with the lowest eigenvalues are the directions in parameter space that are least constrained by the observations. We implement this work within the dolfin-adjoint [3] software package. We derive the MPI-parallel finite element discretisation of the forward, adjoint (1st and 2nd order), and tangent linear models using the high-level differentiation tools available within the FEniCS project. We show results demonstrating the effects of partial observations and poor experimental design on the reliability of the recovered parameters. [1] N. H. Gokhale, P. E. Barbone, and A. A. Oberai, “Solution of the nonlinear elasticity imaging inverse problem: the compressible case,” Inverse Problems, 10.1088/0266-5611/24/4/045010 [2] H. P. Flath, L. C. Wilcox, V. Akçelik, J. Hill, B. van Bloemen Waanders, and O. Ghattas, “Fast Algorithms for Bayesian Uncertainty Quantification in Large-Scale Linear Inverse Problems Based on Low-Rank Partial Hessian Approximations,” SIAM J. Sci. Comput., 10.1137/090780717 [3] P. Farrell, D. Ham, S. Funke, and M. Rognes, “Automated Derivation of the Adjoint of High-Level Transient Finite Element Programs,” SIAM J. Sci. Comput., 10.1137/120873558 [less ▲]

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See detailQuasicontinuum methods for planar beam lattices (abstract)
Beex, Lars UL; Kerfriden, Pierre; Heaney, Claire et al

Scientific Conference (2015, July)

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See detailMulti-scale fracture, model reduction, CAD and image as a model
Bordas, Stéphane UL; Kerfriden, Pierre; Beex, Lars UL et al

Scientific Conference (2015, June 24)

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See detailXFEM with global enrichment for 3D cracks
Agathos, Kostas; Chatzi, Eleni; Bordas, Stéphane UL et al

Scientific Conference (2015, June 17)

We present an extended finite element method (XFEM) based on fixed area enrichment which 1) suppresses the difficulties associated with ill-conditioning, even for "large" enrichment radii; 2) requires 50 ... [more ▼]

We present an extended finite element method (XFEM) based on fixed area enrichment which 1) suppresses the difficulties associated with ill-conditioning, even for "large" enrichment radii; 2) requires 50 times fewer enriched degrees of freedom (for a typical mesh) as the standard XFEM with geometrical enrichment (for the same or better accuracy level); 3) increases the accuracy level of the stress intensity factors and leads to "smooth" stress intensity variations along the crack front. [less ▲]

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See detailAn efficient Computational approach for control of nonlinear transient responses of smart piezoelectric composite plates
Phung-Van, P.; Nguyen, Lieu B.; V. Tran, Loc et al

in International Journal of Non-Linear Mechanics (2015)

An efficient computational approach based on a generalized unconstrained approach in conjunction with isogeometric analysis (IGA) are proposed for dynamic control of smart piezoelectric composite plates ... [more ▼]

An efficient computational approach based on a generalized unconstrained approach in conjunction with isogeometric analysis (IGA) are proposed for dynamic control of smart piezoelectric composite plates. In composite plates, the mechanical displacement field is approximated according to the proposal model using isogeometric elements and the nonlinear transient formulation for plates is formed in the total Lagrange approach based on the von Kármán strains and solved by Newmark time integration. Through the thickness of each piezoelectric layer, the electric potential is assumed linearly. For active control of the piezoelectric composite plates, a close-loop system is used. An optimization procedure using genetic algorithm (GA) is considered to search optimal design for actuator input voltages. Various numerical examples are investigated to show high accuracy and reliability of the proposed method. [less ▲]

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See detailGeometry-Independent Field approximaTion (GIFT) for spline based FEM for Linear Elasticity: a Diffpack implementation
Hossain, Md Naim; Xu, Gang; Bordas, Stéphane UL et al

Scientific Conference (2015, June 01)

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See detailIsogeometric analysis: an overview and computer implementation aspects
Nguyen, Vinh-Phu; Anitescu, Cosmin; Bordas, Stéphane UL et al

in Mathematics and Computers in Simulation (2015)

Isogeometric analysis (IGA) represents a recently developed technology in computational mechanics that offers the possibility of integrating methods for analysis and Computer Aided Design (CAD) into a ... [more ▼]

Isogeometric analysis (IGA) represents a recently developed technology in computational mechanics that offers the possibility of integrating methods for analysis and Computer Aided Design (CAD) into a single, unified process. The implications to practical engineering design scenarios are profound, since the time taken from design to analysis is greatly reduced, leading to dramatic gains in efficiency. In this manuscript, through a self-contained Matlab⃝R implementation, we present an introduction to IGA applied to simple analysis problems and the related computer implementation aspects. Furthermore, implementation of the extended IGA which incorporates enrichment functions through the partition of unity method (PUM) is also presented, where several examples for both two-dimensional and three-dimensional fracture are illustrated. We also describe the use of IGA in the context of strong-form (collocation) formulations, which has been an area of research interest due to the potential for significant efficiency gains offered by these methods. The code which accompanies the present paper can be applied to one, two and three-dimensional problems for linear elasticity, linear elastic fracture mechanics, structural mechanics (beams/plates/shells including large displacements and rotations) and Poisson problems with or without enrichment. The B ́ezier extraction concept that allows the FE analysis to be performed efficiently on T-spline geometries is also incorporated. The article includes a summary of recent trends and developments within the field of IGA. [less ▲]

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See detailControl of Flame Spray Pyrolysis synthesis of Li4Ti5O12: Experimental and Computational study
Tsikourkitoudi, Vasiliki; Gavriliadis, Panagiotis; Bourantas, Georgios UL et al

Poster (2015, May 14)

Lithium titanate (Li4Ti5O12, LTO) is a promising anode material for the next generation of lithium ion batteries. Its physical properties and morphology (which consequently affect its electrochemical ... [more ▼]

Lithium titanate (Li4Ti5O12, LTO) is a promising anode material for the next generation of lithium ion batteries. Its physical properties and morphology (which consequently affect its electrochemical performance) highly depend on its synthesis method. Flame spray pyrolysis (FSP) is an attractive process for the controlled one-step synthesis of functional multicomponent oxides from low cost precursors. The main aim of this study is to control the growth process of LTO by FSP in order to maintain the desired particle properties. LTO nanoparticles of different sizes are synthesized by variation of the FSP processing conditions and characterized accordingly. Numerical simulations based on Population Balance Models are also implemented in order to investigate the evolution of primary and agglomerate particle growth. [less ▲]

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See detailA tutorial on multiple crack growth and intersections with XFEM
Sutula, Danas; Bordas, Stéphane UL

Presentation (2015, May 12)

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See detailThe stable GFEM. Convergence, accuracy and Diffpack implementation
Alves Paladim, Daniel; Natarajan, Sundarajan; Bordas, Stéphane UL et al

Presentation (2015, May 12)

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See detailIsogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory
P., Phung-Van; M., Abdel-Wahab; K.M., Liew et al

in Composite Structures (2015), 123

This paper presents a simple and effective formulation based on isogeometric Analysis (IGA) and higher-order shear deformation theory (HSDT) to investigate the static and dynamic vibration behaviour of ... [more ▼]

This paper presents a simple and effective formulation based on isogeometric Analysis (IGA) and higher-order shear deformation theory (HSDT) to investigate the static and dynamic vibration behaviour of functionally graded carbon nano-reinforced composite plates. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded through the thickness direction according to several linear distributions of the volume fraction of carbon nanotubes. The governing equation is approximated according to the HSDT model using isogeometric elements based on Non-Uniform Rational B-Spline (NURBS) basis functions. This achieves naturally any desired degree of continuity through the choice of the interpolation order, so that the method easily fulfils the C1-continuity requirement of the HSDT model. The accuracy and reliability of the proposed method is verified by comparing its numerical predictions with those of other available numerical approaches. [less ▲]

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See detailAn isogeometric boundary element method for fracture modeling
Peng, Xuan; Atroshchenko, Elena; Bordas, Stéphane UL

Presentation (2015, May)

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See detailHybrid lattice-continuum approach for medical simulations
Bui, Huu Phuoc; Bordas, Stéphane UL

Presentation (2015, May)

For problems in which discontinuities are dominant such as fracture of quasi-brittle materials or cutting of soft tissues, discrete approach is a more suitable than the continuum one since discontinuities ... [more ▼]

For problems in which discontinuities are dominant such as fracture of quasi-brittle materials or cutting of soft tissues, discrete approach is a more suitable than the continuum one since discontinuities can be represented naturally. Lattice models are the good candidate, especially when the mesostructure of the studied material needs to be modeled explicitly. Indeed, the outcome of cutting, tearing, needle insertion and similar operations on soft tissues is significantly affected by the microstructure of the material. In this contribution, a master/slave approach is used in order to couple finite elements and lattice approach in a multi-domain and multiscale framework. [less ▲]

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See detailReduced order methods
Schenone, Elisa UL; Hale, Jack UL; Beex, Lars UL et al

Presentation (2015, April 16)

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See detailHybrid mesh/particle meshless method for geological flows with discontinuous transport properties
Bourantas, Georgios UL; Lavier, Luc; Claus, Susanne et al

Scientific Conference (2015, April 12)

Geodynamic modeling is an important branch of Earth Sciences. Direct observation of geodynamic processes is limited in both time and space, while on the other hand numerical methods are capable of ... [more ▼]

Geodynamic modeling is an important branch of Earth Sciences. Direct observation of geodynamic processes is limited in both time and space, while on the other hand numerical methods are capable of simulating millions of years in a matter of days on a desktop computer. The model equations can be reduced to a set of Partial Differential Equations with possibly discontinuous coefficients, governing mass, momentum and heat transfer over the domain. Some of the major challenges associated with such simulations are (1) geological time scales, which require long (in physical time) simulations using small time steps; (2) the presence of localization zones over which large gradients are present and which are much smaller than the overall physical dimensions of the computational domain and require much more refined discretization than for the rest of the domain, much like in fracture or shear band mechanics. An added difficulty is that such structures in the solution may appear after long periods of stagnant behaviour; (3) the definition of boundary conditions, material parameters and that of a suitable computational domain in terms of size; (4) a posteriori error estimation, sensitivity analysis and discretization adaptivity for the resulting coupled problem, including error propagation between different unknown fields. Consequently, it is arguable that any suitable numerical methods aimed at the solution of such problems on a large scale must be able to (i) provide ease of discretization refinement, including possible partition of unity enrichment; (ii) offer a large stability domain, so that “large” time steps can be chosen; (iii) ease of parallelization and good scalability. Our approach is to rely on “meshless” methods based on a point collocation strategy for the discretization of the set of PDEs. The method is hybrid Eulerian/Lagrangian, which enables to switch easily between stagnant periods and periods of localization. Mass and momentum equations are solved using a meshless point collocation Eulerian method, while energy equation are solved using a set of particles, distributed over the spatial domain, with the solution interpolated back to the Eulerian grid at every time step. This hybrid approach allows for the accurate calculation of fine thermal structures, through the ease of adaptivity offered by the flexibility of the particle method. The approximation space is constructed using the Discretization Correction Particle Strength Exchange (DC PSE) method. The proposed scheme gives the capability of solving flow equations (Stokes flow) in fully irregular geometries while particles, “sprinkled” in the spatial domain, are used to solve convection-diffusion problems avoiding the oscillation produced in the Eulerian approach. The resulting algebraic linear systems were solved using direct solvers. Our hybrid approach can capture sharp variations of stresses and thermal gradients in problems with a strongly variable viscosity and thermal conductivity as demonstrated through various benchmarking test cases such as the development of Rayleigh-Taylor instabilities, viscous heating and flows with non-Newtonian rheology. [less ▲]

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