An algorithm for triangulating smooth three-dimensional domains immersed in universal meshes

We describe an algorithm to recover a boundary-fitting triangulation for a bounded C²-regular domain immersed in a nonconforming background mesh of tetrahedra. The algorithm consists in identifying a polyhedral domain ω_h bounded by facets in the background mesh and morphing ω_h into a boundary-fitting polyhedral approximation Ω_h of Ω. We discuss assumptions on the regularity of the domain, on element sizes and on specific angles in the background mesh that appear to render the algorithm robust. With the distinctive feature of involving just small perturbations of a few elements of the background mesh that are in the vicinity of the immersed boundary, the algorithm is designed to benefit numerical schemes for simulating free and moving boundary problems. In such problems, it is now possible to immerse an evolving geometry in the same background mesh, called a universal mesh, and recover conforming discretizations for it. In particular, the algorithm entirely avoids remeshing-type operations and its complexity scales approximately linearly with the number of elements in the vicinity of the immersed boundary. We include detailed examples examining its performance.     

Three‐dimensional superconvergent gradient recovery on tetrahedral meshes

In this paper, finite element superconvergence phenomenon based on centroidal Voronoi Delaunay tessellations (CVDT) in three‐dimensional space is investigated. The Laplacian operator with the Dirichlet boundary condition is considered. A modified superconvergence patch recovery (MSPR) method is established to overcome the influence of slivers on CVDT meshes. With these two key preconditions, a CVDT mesh and the MSPR, the gradients recovered from the linear finite element solutions have superconvergence in the l₂ norm at nodes of a CVDT mesh for an arbitrary three‐dimensional bounded domain. Numerous numerical examples are presented to demonstrate this superconvergence property and good performance of the MSPR method. Copyright © 2016 John Wiley & Sons, Ltd.     

Universal meshes: A method for triangulating planar curved domains immersed in nonconforming meshes

We introduce a new method to triangulate planar, curved domains that transforms a specific collection of triangles in a background mesh to conform to the boundary. In the process, no new vertices are introduced, and connectivities of triangles are left unaltered. The method relies on a novel way of parameterizing an immersed boundary over a collection of nearby edges with its closest point projection. To guarantee its robustness, we require that the domain be C²‐regular, the background mesh be sufficiently refined near the boundary, and that specific angles in triangles near the boundary be strictly acute. The method can render both straight‐edged and curvilinear triangulations for the immersed domain. The latter includes curved triangles that conform exactly to the immersed boundary, and ones constructed with isoparametric mappings to interpolate the boundary at select points. High‐order finite elements constructed over these curved triangles achieve optimal accuracy, which has customarily proven difficult in numerical schemes that adopt nonconforming meshes. Aside from serving as a quick and simple tool for meshing planar curved domains with complex shapes, the method provides significant advantages for simulating problems with moving boundaries and in numerical schemes that require iterating over the geometry of domains. With no conformity requirements, the same background mesh can be adopted to triangulate a large family of domains immersed in it, including ones realized over several updates during the coarse of simulating problems with moving boundaries. We term such a background mesh as a universal mesh for the family of domains it can be used to triangulate. Universal meshes hence facilitate a framework for finite element calculations over evolving domains while using only fixed background meshes. Furthermore, because the evolving geometry can be approximated with any desired order, numerical solutions can be computed with high‐order accuracy. We present demonstrative examples using universal meshes to simulate the interaction of rigid bodies with Stokesian fluids. Copyright © 2014 John Wiley & Sons, Ltd.     

Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods

We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally convergent polyhedral FEM. We show that the stiffness matrix computed with one subcell SFEM is identical to the consistency term of the virtual element method, irrespective of the topology of the element, as long as the shape functions vary linearly on the boundary. Using this connection, we propose a new stable approach to strain smoothing for polygonal/polyhedral elements where, instead of using sub‐triangulations, we are able to use one single polygonal/polyhedral subcell for each element while maintaining stability. For a similar number of degrees of freedom, the proposed approach is more accurate than the conventional SFEM with triangular subcells. The time to compute the stiffness matrix scales with the in case of the conventional polygonal FEM, while it scales as in the proposed approach. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. Copyright © 2015 John Wiley & Sons, Ltd.     

Randomized low-rank approximation methods for projection-based model order reduction of large nonlinear dynamical problems

Projection-based nonlinear model order reduction (MOR) methods typically make use of a reduced basis to approximate high-dimensional quantities. However, the most popular methods for computing V , eg, through a singular value decomposition of an m × n snapshot matrix, have asymptotic time complexities of and do not scale well as m and n increase. This is problematic for large dynamical problems with many snapshots, eg, in case of explicit integration. In this work, we propose the use of randomized methods for reduced basis computation and nonlinear MOR, which have an asymptotic complexity of only or . We evaluate the suitability of randomized algorithms for nonlinear MOR and compare them to other strategies that have been proposed to mitigate the demanding computing times incurred by large nonlinear models. We analyze the computational complexities of traditional, iterative, incremental, and randomized algorithms and compare the computing times and accuracies for numerical examples. The results indicate that randomized methods exhibit an extremely high level of accuracy in practice, while generally being faster than any other analyzed approach. We conclude that randomized methods are highly suitable for the reduction of large nonlinear problems.     

Nonconforming rotated Q1 tetrahedral element with explicit time stepping for elastodynamics

In this paper, we apply a rotated bilinear tetrahedral element to elastodynamics in . This element performs superior to the constant strain element in bending and, unlike the conforming linear strain tetrahedron, allows for row‐sum lumping of the mass matrix. We study the effect of different choices of approximation (pointwise continuity versus edge average continuity) as well as lumping versus consistent mass in the setting of eigenvibrations. We also use the element in combination with the leapfrog method for time domain computations and make numerical comparisons with the constant strain and linear strain tetrahedra. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust and efficient monolithic fluid‐structure‐interaction solvers

We construct new robust and efficient preconditioned generalized minimal residual solvers for the monolithic linear systems of algebraic equations arising from the finite element discretization and Newton's linearization of the fully coupled fluid–structure interaction system of partial differential equations in the arbitrary Lagrangian–Eulerian formulation. We admit both linear elastic and nonlinear hyperelastic materials in the solid model and cover a large range of flows, for example, water, blood, and air, with highly varying density. The preconditioner is constructed in form of , where , , and are proper approximations to the matrices L, D, and U in the LDU block factorization of the fully coupled system matrix, respectively. The inverse of the corresponding Schur complement is approximated by applying a few cycles of a special class of algebraic multigrid methods to the perturbed fluid sub‐problem, which is obtained by modifying corresponding entries in the original fluid matrix with an explicitly constructed approximation to the exact perturbation coming from the sparse matrix–matrix multiplications. The numerical studies presented impressively demonstrate the robustness and the efficiency of the preconditioner proposed in the paper. Copyright © 2016 John Wiley & Sons, Ltd.     

Subdivision based isogeometric analysis for geometric flows

We present a new isogeometric analysis (IGA) approach based on extended Loop subdivision scheme for solving various geometric flows defined on subdivision surfaces. The studied flows include the second-order, fourth-order, and sixth-order geometric flows, such as averaged mean curvature flow, constant mean curvature flow, and minimal mean-curvature-variation flow, which are generally derived by minimizing the associate energy functionals with $L^2$-gradient flow respectively. The geometric flows are discretized by means of subdivision based IGA, where the finite element space is formulated by the limit form of the extended Loop subdivision for different initial control meshes. The basis functions, consisting of quartic box-splines corresponding to each subdivided control mesh, are utilized to represent the geometry exactly. For the cases of the evolution of open surfaces with any shape boundary, high-order continuous boundary conditions derived from the mixed variational forms of the geometric flows should be implemented to be consistent with the isogeometric concept. For time discretization, we adopt an adaptive semi-implicit Euler scheme. By several numerical experiments, we study the convergence behaviors of the proposed approach for solving the geometric flows with high-order boundary conditions. Moreover, the numerical results also show the accuracy and efficiency of the proposed method.     

Fixed-precision randomized low-rank approximation methods for nonlinear model order reduction of large systems

Many model order reduction (MOR) methods employ a reduced basis to approximate the state variables. For nonlinear models, V is often computed using the snapshot method. The associated low-rank approximation of the snapshot matrix can become very costly as m,n grow larger. Widely used conventional singular value decomposition methods have an asymptotic time complexity of , which often makes them impractical for the reduction of large models with many snapshots. Different methods have been suggested to mitigate this problem, including iterative and incremental approaches. More recently, the use of fast and accurate randomized methods was proposed. However, most work so far has focused on fixed-rank approximations, where rank k is assumed to be known a priori. In case of nonlinear MOR, stating a bound on the precision is usually more appropriate. We extend existing research on randomized fixed-precision algorithms and propose a new heuristic for accelerating reduced basis computation by predicting the rank. Theoretical analysis and numerical results show a good performance of the new algorithms, which can be used for computing a reduced basis from large snapshot matrices, up to a given precision ε.     

Geodesic finite elements on simplicial grids

We introduce geodesic finite elements as a conforming way to discretize partial differential equations for functions v : Ω → M, where Ω is an open subset of and M is a Riemannian manifold. These geodesic finite elements naturally generalize standard first‐order finite elements for Euclidean spaces. They also generalize the geodesic finite elements proposed for d = 1 in a previous publication of the author. Our formulation is equivariant under isometries of M and, hence, preserves objectivity of continuous problem formulations. We concentrate on partial differential equations that can be formulated as minimization problems. Discretization leads to algebraic minimization problems on product manifolds Mⁿ. These can be solved efficiently using a Riemannian trust‐region method. We propose a monotone multigrid method to solve the constrained inner problems with linear multigrid speed. As an example, we numerically compute harmonic maps from a domain in to S². Copyright © 2012 John Wiley & Sons, Ltd.     

A stable nonconforming finite element on hexahedra

A new nonconforming brick element is introduced, which only has 13 DOFs locally and takes as its shape functions space. The vector‐valued version generates, together with a discontinuous approximation, an inf‐sup stable finite element pair of order 2 for the Stokes problem in the energy norm. Copyright © 2016 John Wiley & Sons, Ltd.     

Considerations on a phase-field model for adhesive fracture

A recently proposed phase-field model for cohesive fracture is examined. Previous investigations have shown stress oscillations to occur when using unstructured meshes. It is now shown that the use of nonuniform rational B-splines (NURBS) as basis functions rather than traditional Lagrange polynomials significantly reduces this oscillatory behavior. Moreover, there is no effect on the global structural behavior, as evidenced through load-displacement curves. The phase-field model imposes restrictions on the interpolation order of the NURBS used for the three different fields: displacement, phase field, and crack opening. This holds within the Bézier element, but also at the boundaries, where a reduction to ${𝒞}^0$-continuity yields optimal results. Application to a range of cases, including debonding of a hard fiber embedded in a soft matrix, illustrates the potential of the cohesive phase-field model.     

A second‐order decoupled implicit/explicit method of the 3D primitive equations of ocean II: finite element spatial discretization

A fully discrete second‐order decoupled implicit/explicit method is proposed for solving 3D primitive equations of ocean in the case of Dirichlet boundary conditions on the side, where a second‐order decoupled implicit/explicit scheme is used for time discretization, and a finite element method based on the P₁(P₁) ? P₁?P₁(P₁) elements for velocity, pressure and density is used for spatial discretization of these primitive equations. Optimal H¹?L²?H¹ error estimates for numerical solution and an optimal L² error estimate for are established under the convergence condition of 0 < h≤β₁,0 < τ≤β₂, and τ≤β₃h for some positive constants β₁,β₂, and β₃. Furthermore, numerical computations show that the H¹?L²?H¹ convergence rate for numerical solution is of O(h + τ²) and an L² convergence rate for is O(h²+τ²) with the assumed convergence condition, where h is a mesh size and τ is a time step size. More practical calculations are performed as a further validation of the numerical method. Copyright © 2016 John Wiley & Sons, Ltd.     

Forced vibrations of a turbine blade undergoing regularized unilateral contact conditions through the wavelet balance method

The method of weighted residuals can efficiently enforce time‐periodic solutions of flexible structures experiencing unilateral contact. The harmonic balance method (HBM) based on Fourier expansion of the sought solution is a common formulation, although wavelet bases that can sparsely define nonsmooth solutions may be superior. This hypothesis is investigated using a full three‐dimensional blade with unilateral contact conditions on a set of N_c discrete contact points located at its tip. The unilateral contact conditions are first regularized, and a distributional formulation in time is introduced, allowing trial functions to properly approximate in the time domain the solution to the governing equations. The mixed wavelet Petrov–Galerkin solutions are found to yield consistent or better results than HBM, with higher convergence rates and seemingly more accurate contact force prediction. Copyright © 2014 John Wiley & Sons, Ltd.     

New benchmark problems for verification of the curve-to-surface contact algorithm based on the generalized Euler–Eytelwein problem

Development of the numerical contact algorithms for finite element method usually concerns convergence, mesh dependency, etc. Verification of the numerical contact algorithm usually includes only a few cases due to a limited number of available analytic solutions (e.g., the Hertz solution for cylindrical surfaces). The solution of the generalized Euler–Eytelwein, or the belt friction problem is a stand alone task, recently formulated for a rope laying in sliding equilibrium on an arbitrary surface, opens up to a new set of benchmark problems for the verification of rope/beam to surface/solid contact algorithms. Not only a pulling forces ratio ${\displaystyle \frac{T}{T_0}}$, but also the position of a curve on a arbitrary rigid surface withstanding the motion in dragging direction should be verified. Particular situations possessing a closed form solution for ropes and rigid surfaces are analyzed. The verification study is performed employing the specially developed Solid-Beam finite element with both linear and $C1$-continuous approximations together with the Curve-to-Solid Beam (CTSB) contact algorithm and exemplary employing commercial finite element software. A crucial problem of "contact locking" in contact elements showing stiff behavior despite the good convergence is identified. This problem is resolved within the developed CTSB contact element.     

An isogeometric analysis approach to gradient‐dependent plasticity

Gradient‐dependent plasticity can be used to achieve mesh‐objective results upon loss of well‐posedness of the initial/boundary value problem because of the introduction of strain softening, non‐associated flow, and geometric nonlinearity. A prominent class of gradient plasticity models considers a dependence of the yield strength on the Laplacian of the hardening parameter, usually an invariant of the plastic strain tensor. This inclusion causes the consistency condition to become a partial differential equation, in addition to the momentum balance. At the internal moving boundary, one has to impose appropriate boundary conditions on the hardening parameter or, equivalently, on the plastic multiplier. This internal boundary condition can be enforced without tracking the elastic‐plastic boundary by requiring ‐continuity with respect to the plastic multiplier. In this contribution, this continuity has been achieved by using nonuniform rational B‐splines as shape functions both for the plastic multiplier and for the displacements. One advantage of this isogeometric analysis approach is that the displacements can be interpolated one order higher, making it consistent with the interpolation of the plastic multiplier. This is different from previous approaches, which have been exploited. The regularising effect of gradient plasticity is shown for 1‐ and 2‐dimensional boundary value problems.     

A shape optimization approach for simulating contact of elastic membranes with rigid obstacles

The obstacle problem consists in computing equilibrium shapes of elastic membranes in contact with rigid obstacles. In addition to the displacement u of the membrane, the interface Γ on the membrane demarcating the region in contact with the obstacle is also an unknown and plays the role of a free boundary. Numerical methods that simulate obstacle problems as variational inequalities share the unifying feature of first computing membrane displacements and then deducing the location of the free boundary a posteriori. We present a shape optimization-based approach here that inverts this paradigm by considering the free boundary to be the primary unknown and compute it as the minimizer of a certain shape functional using a gradient descent algorithm. In a nutshell, we compute Γ then u, and not u then Γ. Our approach proffers clear algorithmic advantages. Unilateral contact constraints on displacements, which render traditional approaches into expensive quadratic programs, appear only as Dirichlet boundary conditions along the free boundary. Displacements of the membrane need to be approximated only over the noncoincidence set, thereby rendering smaller discrete problems to be resolved. The issue of suboptimal convergence of finite element solutions stemming from the reduced regularity of displacements across the free boundary is naturally circumvented. Most importantly perhaps, our numerical experiments reveal that the free boundary can be approximated to within distances that are two orders of magnitude smaller than the mesh size used for spatial discretization. The success of the proposed algorithm relies on a confluence of factors- choosing a suitable shape functional, representing free boundary iterates with smooth implicit functions, an ansatz for the velocity of the free boundary that helps realize a gradient descent scheme and triangulating evolving domains with universal meshes. We discuss these aspects in detail and present numerous examples examining the performance of the algorithm.     

Two‐sided Grassmann manifold algorithm for optimal model reduction

We consider an optimal model reduction problem for large‐scale dynamical systems. The problem is formulated as a minimization problem over Grassmann manifold with two variables. This formulation allows us to develop a two‐sided Grassmann manifold algorithm, which is numerically efficient and suitable for the reduction of large‐scale systems. The resulting reduced system preserves the stability of the original system. Numerical examples are presented to show that the proposed algorithm is computationally efficient and robust with respect to the selection of initial projection matrices. Copyright © 2015 John Wiley & Sons, Ltd.     

Extended finite element method for plastic limit load computation of cracked structures

The extended finite element method is extended to allow computation of the limit load of cracked structures. In the paper, it is demonstrated that the linear elastic tip enrichment basis with and without radial term may be used in the framework of limit analysis, but the six‐function enrichment basis based on the well‐known Hutchinson–Rice–Rosengren asymptotic fields appears to be the best. The discrete kinematic formulation is cast in the form of a second‐order cone problem, which can be solved using highly efficient interior‐point solvers. Finally, the proposed numerical procedure is applied to various benchmark problems, showing that the present results are in good agreement with those in the literature. Copyright © 2015 John Wiley & Sons, Ltd.