Bishnu Lamichhane

University of Newcastle Australia · School of Mathematical and Physical Sciences

We develop a family of mixed finite element methods for a model of nonlinear poroelasticity where, thanks to a rewriting of the constitutive equations, the permeability depends on the total poroelastic stress and on the fluid pressure and therefore we can use the Hellinger-Reissner principle with weakly imposed stress symmetry for Biot's equations....

We propose four-field and five-field Hu–Washizu-type mixed formulations for nonlinear poroelasticity – a coupled fluid diffusion and solid deformation process – considering that the permeability depends on a linear combination between fluid pressure and dilation. As the determination of the physical strains is necessary, the first formulation is wr...

We propose four-field and five-field Hu--Washizu-type mixed formulations for nonlinear poroelasticity -- a coupled fluid diffusion and solid deformation process -- considering that the permeability depends on a linear combination between fluid pressure and dilation. As the determination of the physical strains is necessary, the first formulation is...

Introduction: In this study, we aimed to investigate the feasibility of gadoxetate low-temporal resolution (LTR) DCE-MRI for voxel-based hepatic extraction fraction (HEF) quantification for liver sparing radiotherapy using a deconvolution analysis (DA) method. Methods: The accuracy and consistency of the deconvolution implementation in estimatin...

Purpose Calibration of a radiotherapy electronic portal imaging device (EPID) using the pixel‐sensitivity‐map (PSM) in place of the flood field correction improves the utility of the EPID for quality assurance applications. Multiple methods are available for determining the PSM and this study provides an evaluation to inform on which is superior....

Purpose: The EPID PSM is a useful EPID calibration method for QA applications. The dependence of the EPID PSM on the photon beam used to acquire it has been investigated in this study for the four available PSM methods. The aim is to inform upon the viability of applying a single PSM for all available photon beams to simplify PSM implementation an...

A solution method is developed for a linear model of ice shelf flexural vibrations in response to ocean waves, in which the ice shelf thickness and seabed beneath the ice shelf vary over distance, and the ice shelf/sub–ice-shelf cavity are connected to the open ocean. The method combines a decomposition of the ice shelf displacement profile at a pr...

We compare a recently proposed multivariate spline based on mixed partial derivatives with two other standard splines for the scattered data smoothing problem. The splines are defined as the minimiser of a penalised least squares functional. The penalties are based on partial differential operators, and are integrated using the finite element metho...

The numerical approximation of hyperelasticity must address nonlinear constitutive laws, geometric nonlinearities associated with large strains and deformations, the imposition of the incompressibility of the solid, and the solution of large linear systems arising from the discretisation of 3D problems in complex geometries. We adapt the three-fiel...

The main objective of the current work is to determine meshless methods using the radial basis function (rbf) approach to estimate the elastic strain field from energy-resolved neutron imaging. To this end, we first discretize the longitudinal ray transformation with rbf methods to give us an unconstrained optimization problem. This discretization...

We use a three‐field mixed formulation of the Poisson equation to develop a mixed finite element method using Raviart–Thomas elements. We use a locally constructed biorthogonal system for Raviart–Thomas finite elements to improve the computational efficiency of the approach. We analyze the existence, uniqueness and stability of the discrete problem...

The motion of a flexible elastic plate under wave action is simulated, and the well–known phenomena of overwash is investigated. The fluid motion is modelled by smoothed particle hydrodynamics, a mesh-free solution method which, while computationally demanding, is flexible and able to simulate complex fluid flows. The freely floating plate is model...

We present a mixed finite element method for the elasticity problem. We expand the standard Hu–Washizu formulation to include a pressure unknown and its Lagrange multiplier. By doing so, we derive a five-field formulation. We apply a biorthogonal system that leads to an efficient numerical formulation. We address the coercivity problem by adding a...

A mathematical model for predicting the vibrations of ice-shelves based on linear elasticity for the ice-shelf motion and potential flow for the fluid motion is developed. No simplifying assumptions such as the thinness of the ice-shelf or the shallowness of the fluid are made. The ice-shelf is modelled as a two-dimensional elastic body of an arbit...

A novel pulsed neutron imaging technique based on the finite element method is used to reconstruct the residual strain within a polycrystalline material from Bragg edge strain images. This technique offers the possibility of a nondestructive analysis of strain fields with a high spatial resolution. The finite element approach used to reconstruct th...

We compare a recently proposed multivariate spline based on mixed partial derivatives with two other standard splines for the scattered data smoothing problem. The splines are defined as the minimiser of a penalised least squares functional. The penalties are based on partial differentiation operators, and are integrated using the finite element me...

The tomographic reconstruction of an object embedded within gelatin has been undertaken applying the Radon transform to visible light scattering recorded by a fast optical scanner, allowing a successful reconstruction of the object.

The solution to the problem of the vibration of an ice shelf of constant thickness is calculated using the eigenfunction matching method in water of finite depth, and accounting for the draught of the shelf.The eigenfunction matching solution is validated against a solution found using the finite element method. The finite-depth solution is careful...

In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is...

The virtual element method is an extension of the finite element method on polygonal meshes. The virtual element basis functions are generally unknown inside an element and suitable projections of the basis functions onto polynomial spaces are used to construct the elemental stiffness and mass matrices. We present a gradient recovery method based o...

In this paper, we consider an optimal control problem governed by elliptic differential equations posed in a three-field formulation. Using the gradient as a new unknown we write a weak equation for the gradient using a Lagrange multiplier. We use a biorthogonal system to discretise the gradient, which leads to a very efficient numerical scheme. A...

The frequency-domain and time-domain response of a floating ice shelf to wave forcing are calculated using the finite element method. The boundary conditions at the front of the ice shelf, coupling it to the surrounding fluid, are written as a special non-local linear operator with forcing. This operator allows the computational domain to be restri...

We consider higher order finite element discretizations of a nonlinear variational inequality formulation arising from an obstacle problem with the p-Laplacian differential operator for p∈(1,∞). We prove an a priori error estimate and convergence rates with respect to the mesh size h and in the polynomial degree q under assumed regularity. Moreover...

Beam steering is the process of calibrating the angle and translational position with which a linear accelerator's (linac's) electron beam strikes the x‐ray target with respect to the collimator rotation axis. The shape of the dose profile is highly dependent on accurate beam steering and is essential for ensuring correct delivery of the radiothera...

We consider a mixed finite element method for an obstacle problem with the p-Laplace differential operator for p ∈ (1 ,∞), where the obstacle condition is imposed by using a Lagrange multiplier. In the discrete setting the Lagrange multiplier basis forms a biorthogonal system with the standard finite element basis so that the variational inequality...

In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is...

The inversion of geophysical potential field data can be formulated as an optimization problem with a constraint in the form of a partial differential equation (PDE). It is common practice, if possible, to provide an analytical solution for the forward problem and to reduce the problem to a finite dimensional optimization problem. In an alternative...

We modify a three-field formulation of the Poisson problem with Nitsche approach for approximating Dirichlet boundary conditions. Nitsche approach allows us to weakly impose Dirichlet boundary condition but still preserves the optimal convergence. We use the biorthogonal system for efficient numerical computation and introduce a stabilisation term...

We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to formulate our saddle point problem and the finite element method. The new formulation allows us to use the $H^1$-con...

We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to formulate our saddle point problem and the finite element method. The new formulation allows us to use the $H^1$-con...

The Douglas-Rachford method has been employed successfully to solve many kinds of non-convex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary valued problem (BVP)...

The Douglas-Rachford method has been employed successfully to solve many kinds of non-convex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary valued problem (BVP)...

An initial-boundary value problem for a time-fractional diffusion equation is discretized in space, using continuous piecewise-linear finite elements on a domain with a re-entrant corner. Known error bounds for the case of a convex domain break down, because the associated Poisson equation is no longer $H^{2}$ -regular. In particular, the method is...

We consider a new three-field formulation of the biharmonic problem. The solution, the gradient and the Lagrange multiplier are the three unknowns in the formulation. Adding a stabilization term in the discrete setting we can use the standard Lagrange finite element to discretize the solution, whereas we use the Raviart-Thomas finite element to dis...

An initial-boundary value problem for the time-fractional diffusion equation is discretized in space using continuous piecewise-linear finite elements on a polygonal domain with a re-entrant corner. Known error bounds for the case of a convex polygon break down because the associated Poisson equation is no longer H 2 - regular. In particular, the m...

We introduce a new minimisation principle for Poisson equation using two variables: the solution and the gradient of the solution. This principle allows us to use any conforming finite element spaces for both variables, where the finite element spaces do not need to satisfy a so-called inf-sup condition. A numerical example demonstrates the superio...

We consider a quadrilateral 'mini' finite element for approximating the solution of Stokes equations using a quadrilateral mesh. We use the standard bilinear finite element space enriched with element-wise defined bubble functions for the velocity and the standard bilinear finite element space for the pressure space. With a simple modification of t...

The thin plate spline method is a widely used data fitting technique as it has the ability to smooth noisy data. Here we consider a mixed finite element discretisation of the thin plate spline. By using mixed finite elements the formulation can be defined in-terms of relatively simple stencils, thus resulting in a system that is sparse and whose si...

A gradient recovery operator based on projecting the discrete gradient onto the standard finite element space is considered. We use an oblique projection, where the test and trial spaces are different, and the bases of these two spaces form a biorthogonal system. Biorthogonality allows efficient computation of the recovery operator.We analyse the a...

We present a stabilized finite element method for a mixed formulation of elasticity equations using the lowest order Crouzeix–Raviart element. The stabilization is done to satisfy Korn’s inequality and is based on an oblique projection operator.

We present a new multivariate spline using mixed partial derivatives. We show the existence and uniqueness of the proposed multivariate spline problem, and propose a simple finite element approximation.

We present a finite element method for nearly incompressible elasticity using a mixed formulation of linear elasticity in the displacement–pressure form. The idea of stabilization of an equal order interpolation for Stokes equations is combined with biorthogonality to get rid of the bubble functions. A Petrov–Galerkin formulation for the pressure e...

A new non-conforming finite element method is proposed for the approximation of the biharmonic equation with clamped boundary condition. The new formulation is based on a gradient recovery operator. Optimal a priori error estimates are proved for the proposed approach. The approach is also extended to cover a singularly perturbed problem.

The Gradient Scheme framework provides a unified analysis setting for many different families of numerical methods for diffusion equations. We show in this paper that the Gradient Scheme framework can be adapted to elasticity equations, and provides error estimates for linear elasticity and convergence results for non-linear elasticity. We also est...

We present a finite element method for Stokes equations using the Crouzeix-Raviart element for the velocity and the continuous linear element for the pressure. We show that the inf-sup condition is satisfied for this pair. Two numerical experiments are presented to support the theoretical results.

We present a simple finite element method for the discretization of Reissner--Mindlin plate equations. The finite element method is based on using the nonconforming Crouzeix-Raviart finite element space for the transverse displacement, and the standard linear finite element space for the rotation of the transverse normal vector. We also present two...

We consider a mixed finite element method for approximating the solution of nearly incompressible elasticity and Stokes equations. The finite element method is based on quadrilateral and hexahedral triangulation using primal and dual meshes. We use the standard bilinear and trilinear finite element space enriched with element-wise defined bubble fu...

We present two simple finite element methods for the discretization of Reissner-Mindlin plate equations with clamped boundary condition. These finite element methods are based on discrete Lagrange multiplier spaces from mortar finite element techniques. We prove optimal a priori error estimates for both methods. The first approach is based on a so-...

We consider a mixed finite element method based on simplicial triangulations for a three-field formulation of linear elasticity. The three-field formulation is based on three unknowns: displacement, stress and strain. In order to obtain an efficient discretization scheme, we use a pair of finite element bases forming a biorthogonal system for the s...

We introduce two three-field mixed formulations for the Poisson equation and propose finite element methods for their approximation. Both mixed formulations are obtained by introducing a weak equation for the gradient of the solution by means of a Lagrange multiplier space. Two efficient numerical schemes are proposed based on using a pair of bases...

We present a new finite element method for Darcy-Stokes-Brinkman equations using primal and dual meshes for the velocity and the pressure, respectively. Using an orthogonal basis for the discrete space for the pressure, we use an efficiently computable stabilization to obtain a uniform convergence of the finite element approximation for both limiti...

We present a symmetric version of the nonsymmetric mixed finite element method presented in (Lamichhane, ANZIAM J 50 (2008), C324–C338) for nearly incompressible elasticity. The displacement–pressure formulation of linear elasticity is discretized using a Petrov–Galerkin discretization for the pressure equation in (Lamichhane, ANZIAM J 50 (2008), C...

We propose a stabilized finite element method for the approximation of the biharmonic equation with a clamped boundary condition. The mixed formulation of the biharmonic equation is obtained by introducing the gradient of the solution and a Lagrange multiplier as new unknowns. Working with a pair of bases forming a biorthogonal system, we can easil...

We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. The method is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation on...

We present a construction of a gradient recovery operator based on an oblique projection, where the basis functions of two involved spaces satisfy a condition of biorthogonality. The biorthogonality condition guarantees that the recovery operator is local.

Thin-plate splines are a well established technique for the inter-polation and smoothing of scattered data. However, the traditional formulation of the method leads to large, dense and often ill-conditioned matrices, which reduces its applicability in practice. We present a new mixed finite element formulation based on the ideas behind the mortar f...

We present a stabilized finite element method for the Hu–Washizu formulation of linear elasticity based on simplicial meshes leading to the stabilized nodal strain formulation or node-based uniform strain elements. We show that the finite element approximation converges uniformly to the exact solution for the nearly incompressible case.

We present a finite element method for non-linear and nearly incompressible elasticity. The formulation is based on Petrov–Galerkin discretization for the pressure and is closely related to the average nodal pressure formulation presented earlier in the context of incompressible and nearly incompressible dynamic explicit applications (Commun. Numer...

The finite element method has become a very powerful and popular tool to solve boundary value problems coming from science and engineering. Here, we consider a scattered data fitting method based on the finite element method and apply the method to remove the mixture of Gaussian and impulsive noise from an image. Numerical results show the performa...

We consider a heat transfer problem with sliding bodies, where heat is generated on the interface due to friction. Neglecting the mechanical part, we assume that the pressure on the contact interface is a known function. Using mortar techniques with Lagrange multipliers, we show existence and uniqueness of the solution in the continuous setting. Mo...

The thin plate spline is a popular tool for the interpolation and smoothing of scattered data. In this paper we propose a novel stabilized mixed finite element method for the discretization of thin plate splines. The mixed formulation is obtained by introducing the gradient of the smoother as an additional unknown. Working with a pair of bases for...

We consider the coupling of compressible and nearly incompressible materials within the frame-work of mortar methods. Taking into account the locking effect, we use a suitable discretization for the nearly incompressible material and work with a standard conforming discretization elsewhere. The coupling of different discretization schemes in differ...

A Petrov-Galerkin scheme in a saddle point formulation gives rise to a non-symmetric saddle point problem. This article considers a non-symmetric saddle point problem with a penalty parameter. A mixed finite element method for linear elasticity based on a Petrov-Galerkin formulation is then analyzed within the framework of the non-symmetric saddle...

We consider finite-element methods based on simplices to solve the problem of nearly incompressible elasticity. Two different approaches based, respectively, on dual meshes and dual bases are presented, where in both approaches pressure is discontinuous and can be statically condensed out from the system. These novel approaches lead to displacement...

The role of sparse representations in the context of structured noise filtering is discussed. A strategy, especially conceived so as to address problems of an ill posed nature, is presented. The proposed approach revises and extends the Oblique Matching Pursuit technique. It is shown that, by working with an orthogonal projection of the signal to b...

The uniform convergence of finite element approximations based on a modified Hu–Washizu formulation for the nearly incompressible linear elasticity is analyzed. We show the optimal and robust convergence of the displacement-based discrete formulation in the nearly incompressible case with the choice of approximations based on quadrilateral and hexa...

We construct locally supported basis functions which are biorthog- onal to conforming nodal nite element basis functions of order p in one di- mension. In contrast to earlier approaches, these basis functions have the same support as the nodal nite element basis functions and reproduce the conform- ing nite element space of order p 1. Working with...

The classical Hu–Washizu mixed formulation for plane problems in elasticity is examined afresh, with the emphasis on behavior in the incompressible limit. The classical continuous problem is embedded in a family of Hu–Washizu problems parametrized by a scalar α for which a = l/ m\alpha = \lambda \big/ \mu corresponds to the classical formulation, w...

The relationship of the Hu–Washizu mixed formulation to other mixed and enhanced formulations is examined in detail, in the context of linear elasticity, with a view to presenting a unified framework for such formulations. The Hu–Washizu formulation is considered in both its classical form and in a modified form that is suited to establishing well-...

The numerical approximation of partial differential equations coming from physical and engineering modeling is often a challenging task. Most often these partial differential equations are discretized with finite elements and can be solved by modern super-computers. Working with different discretization techniques in different subdomains or indepen...

Mortar methods with dual Lagrange multiplier bases provide a flexible, efficient and optimal way to couple different discretization schemes or nonmatching triangulations. Here, we generalize the concept of dual Lagrange multiplier bases by relaxing the condition that the trace space of the approximation space at the slave side with zero boundary co...

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We consider mortar techniques with dual Lagrange multiplier spaces to couple different discretization schemes. It is well known that the discretization error for linear mortar finite elements in the energy norm is of order h. H...

Mortar techniques provide a flexible tool for the coupling of different discretization schemes or triangulations. Here, we consider interface problems within the framework of mortar finite element methods. We start with a saddle point formulation and show that the interface conditions enter into the right-hand side. Using dual Lagrange multipliers,...

Domain decomposition techniques provide a flexible tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements in 3D with different Lagrange multiplier spaces. In particular, we focus on Lagrange multiplier spaces which yield optimal discretization schemes and a locally s...

Domain decomposition techniques provide a flexible tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements in 3D with different Lagrange multiplier spaces. In particular, we focus on Lagrange multiplier spaces which yield optimal discretization schemes and a locally s...

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements. In particular, we focus on dual Lagrange multiplier spaces. These non-standard Lagrange multiplier spaces yield optimal discretization schemes and a locally sup...

For many inverse problems which arise in contributing to real-world decision-making, such as formulating policy objectives for freshwater fish health, only an indicative understanding of the global structure of the solution is all that is required for the associated decision-support. Examples include: (i) Situations where only some linear functiona...

In this chapter we present the relevant mathematical background to address two well defined signal and image processing problems. Namely, the problem of struc-tured noise filtering and the problem of interpolation of missing data. The former is addressed by recourse to oblique projection based techniques whilst the latter, which can be considered e...