A Bayesian inversion approach to recovering material parameters in hyperelastic solids using dolfin-adjoint

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 ▲]

Hyperelastic Elastography in a Large-Scale Bayesian Inversion Setting

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 ▲]

Reduced order methods

Presentation (2015, April 16)

Isogeometric locking-free plate element: a simple first order shear deformation theory for functionally graded plates

in Composite Structures (2014), 118

An effective, simple, robust and locking-free plate formulation is proposed to analyze the static bending, buckling, and free vibration of homogeneous and functionally graded plates. The simple first ... [more ▼]

An effective, simple, robust and locking-free plate formulation is proposed to analyze the static bending, buckling, and free vibration of homogeneous and functionally graded plates. The simple first-order shear deformation theory (S-FSDT), which was recently presented in Thai and Choi (2013) [11], is naturally free from shear-locking and captures the physics of the shear-deformation effect present in the original FSDT, whilst also being less computationally expensive due to having fewer unknowns. The S-FSDT requires C1-continuity that is simple to satisfy with the inherent high-order continuity of the non-uniform rational B-spline (NURBS) basis functions, which we use in the framework of isogeometric analysis (IGA). Numerical examples are solved and the results are compared with reference solutions to confirm the accuracy of the proposed method. Furthermore, the effects of boundary conditions, gradient index, and geometric shape on the mechanical response of functionally graded plates are investigated. [less ▲]

Cardiff/Luxembourg Computational Mechanics Research Group

Poster (2014, November)

Discrete Multiscale Modelling and Future Research Plans concerning Metals (presentation)

Presentation (2014, October 14)

Meshfree methods for shear-deformable structures based on mixed weak forms

Scientific Conference (2014, July 24)

Similarly to the finite element method, meshfree methods must be carefully designed to overcome the shear-locking problem when discretising the shear-deformable structural theories. Many successful ... [more ▼]

Similarly to the finite element method, meshfree methods must be carefully designed to overcome the shear-locking problem when discretising the shear-deformable structural theories. Many successful treatments of shear-locking in the finite element literature are constructed through the application of a mixed variational form, where the shear stress is treated as an independent variational quantity in addition to the usual displacements. Because of its sound mathematical underpinnings this is the methodology I have chosen to solve the shear-locking problem when using meshfree basis functions. In this talk I will discuss the mathematical origins of the shear-locking problem and the applicability of the celebrated LBB stability condition for designing well-behaved mixed meshfree approximation schemes. I will show results from two new formulations that demonstrate the effectiveness of this approach. The first method is a meshfree formulation for the Timoshenko beam problem that converges to a classic inf-sup stable finite element method when using Maximum- Entropy basis functions. The second method is a generalised displacement meshfree method for the Reissner- Mindlin problem where the shear stress is eliminated prior to the solution of the linear system using a local patch-projection technique, resulting in a linear system expressed in terms of the original displacement unknowns only. Stability is ensured by using a stabilised weak form which is necessary due to the loss of kernel coercivity for the Reissner-Mindlin problem. [less ▲]

Meshfree volume-averaged nodal projection methods for incompressible media problems

Scientific Conference (2014, July 21)

Stress analysis, damage tolerance assessment and shape optimisation without meshing

Poster (2014, June 24)

Reducing the Mesh-burden and Computational Expense in Multi-scale Free Boundary Engineering Problems

Presentation (2014, May 12)

We present recent results aiming at affording faster and error-controlled simulations of multi scale phenomena including fracture of heterogeneous materials and cutting of biological tissue. In a second ... [more ▼]

We present recent results aiming at affording faster and error-controlled simulations of multi scale phenomena including fracture of heterogeneous materials and cutting of biological tissue. In a second part, we describe methodologies to isolate the user from the burden of mesh generation and regeneration as moving boundaries evolve. Results include advances in implicit boundary finite elements, (enriched) isogeometric boundary elements and extended finite element methods for multi-crack propagation. ABOUT THE PRESENTER In 1999, Stéphane Bordas joined a joint graduate programme of the French Institute of Technology (Ecole Spéciale des Travaux Publics) and the American Northwestern University. In 2003, he graduated in Theoretical and Applied Mechanics with a PhD from Northwestern University. Between 2003 and 2006, he was at the Laboratory of Structural and Continuum Mechanics at the Swiss Federal Institute of Technology in Lausanne, Switzerland. In 2006, he became permanent lecturer at Glasgow University’s Civil Engineering Department. Stéphane joined the Computational Mechanics team at Cardiff University in September 2009, as a Professor in Computational Mechanics and directed the institute of Mechanics and Advanced Materials from October 2010 to November 2013. He is the Editor of the book series “Advances in Applied Mechanics” since July 2013. In November 2013, he joined the University of Luxembourg as a Professor in Computational Mechanics. The main axes of his research team include (1) free boundary problems and problems involving complex geometries, in particular moving boundaries and (2) ‘a posteriori’ discretisation and model error control, rationalisation of the computational expense. Stéphane’s keen interest is to actively participate in innovation, technological transfer as well as software tool generation. This has been done through a number of joint ventures with various industrial partners (Bosch GmbH, Cenaero, inuTech GmbH, Siemens-LMS, Soitec SA) and the release of open-source software. In 2012, Stéphane was awarded an ERC Starting Independent Research Grant (RealTcut), to address the need for surgical simulators with a computational mechanics angle with a focus on the multi-scale simulation of cutting of heterogeneous materials in real-time. [less ▲]