ESOLID—a system for exact boundary evaluation

We present a system, ESOLID, that performs exact boundary evaluation of low-degree curved solids in reasonable amounts of time. ESOLID performs accurate Boolean operations using exact representations and exact computations throughout. The demands of exact computation require a different set of algorithms and efficiency improvements than those found in a traditional inexact floating point based modeler. We describe the system architecture, representations, and issues in implementing the algorithms. We also describe a number of techniques that increase the efficiency of the system based on lazy evaluation, use of floating point filters, arbitrary floating point arithmetic with error bounds, and lower dimensional formulation of subproblems. ESOLID has been used for boundary evaluation of many complex solids. These include both synthetic datasets and parts of a Bradley Fighting Vehicle designed using the BRL-CAD solid modeling system. It is shown that ESOLID can correctly evaluate the boundary of solids that are very hard to compute using a fixed-precision floating point modeler. In terms of performance, it is about an order of magnitude slower as compared to a floating point boundary evaluation system on most cases

Exact polynomial system solving for robust geometric computation

I describe an exact method for computing roots of a system of multivariate polynomials with rational coefficients, called the rational univariate reduction. This method enables performance of exact algebraic computation of coordinates of the roots of polynomials. In computational geometry, curves, surfaces and points are described as polynomials and their intersections. Thus, exact computation of the roots of polynomials allows the development and implementation of robust geometric algorithms. I describe applications in robust geometric modeling. In particular, I show a new method, called numerical perturbation scheme, that can be used successfully to detect and handle degenerate configurations appearing in boundary evaluation problems. I develop a derandomized version of the algorithm for computing the rational univariate reduction for a square system of multivariate polynomials and a new algorithm for a non-square system. I show how to perform exact computation over algebraic points obtained by the rational univariate reduction. I give a formal description of numerical perturbation scheme and its implementation

A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method

This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solver (or its arc length variant) have been reported not to converge (refer to References Wang et al. (2014), Lahuerta et al. (2013) and Liu et al. (2017)). In this paper, the new TO stabilisation technique proposed in Ortigosa et al. (2019) in the context of level-set TO, initially devised to alleviate numerical instabilities inherent to level-set TO, is extended for the TO by means of the SIMP method. The success of the methodology rests on the combination of two distinct key ingredients. First, the nonlinear equilibrium equations of motion for intermediate TO design stages are solved in a non-exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is locally stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. This solution strategy is shown to be extremely robust in the context of density-based TO, where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents, the latter classically defined by means of a fictitious strain energy. The robustness and applicability of this TO methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the final designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton–Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solver (or its arc length variant) have been reported not to converge (refer to References Wang et al. (2014), Lahuerta et al. (2013) and Liu et al. (2017)). In this paper, the new TO stabilisation technique proposed in Ortigosa et al. (2019) in the context of level-set TO, initially devised to alleviate numerical instabilities inherent to level-set TO, is extended for the TO by means of the SIMP method. The success of the methodology rests on the combination of two distinct key ingredients. First, the nonlinear equilibrium equations of motion for intermediate TO design stages are solved in a non-exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is locally stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. This solution strategy is shown to be extremely robust in the context of density-based TO, where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents, the latter classically defined by means of a fictitious strain energy. The robustness and applicability of this TO methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the final designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton–Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10 -12

EFFICIENT POLYNOMIAL ROOT ISOLATION APPLIED TO COMPUTATIONAL GEOMETRY

Over the last several years, the field of polynomial root isolation has been rapidly improving, but the computational geometry applications have been somewhat unexplored. Here, we have an implementation of a curve intersection engine that showcases the current state-of-the-art in root isolation. The engine is capable of taking two implicitly defined curves and locating their intersection points within some required accuracy. From this work, we can clearly see that root isolation is no longer a significant speed issue in computational geometry. The next issue is really speed of the resultant computation used for variable eliminatio

Application of Engineered Porosity and Modified Effective Moduli to the Design of Orthopaedic Implants

Commercially available orthopaedic implants have a bending stiffness (flexural rigidity) that is at least 10 times greater than cortical bone. Effects of this stiffness mismatch have been extensively studied relative to total hip arthroplasty (THA). Clinical experience with THA has shown that stiffness mismatch is the primary cause of accelerated bone resorption due to the stress shielding, resulting in sub-optimal bone loading, aseptic loosening and inadequate bone support for a future revision implant. Attempts to incorporate design features that reduce the flexural rigidity of implants have yielded inconsistent results or failures due to biomaterial incompatibilities and practical manufacturing complications. The recent development of additive manufacturing (AM) processes allow the fabrication of closed-cell porous Ti or CoCr microstructures as a practical means of fabrication while reducing implant stiffness. The use of engineered porosity to modify flexural rigidity requires an ability to predict moduli from microstructural parameters. The literature is replete with different formulas which are often contradictory; existing equations relating porosity to effective moduli are generally interpretive and not predictive. This study applied finite element methods to three-dimensional porous structures with different arrangements of spheroidal voids. The resulting data show that the effective Young\u27s modulus varies linearly with &psi, the ratio of pore radius to center-to-center dimension, for a porosity range of 20 to 50%. In addition, the arrangement of spherical voids was found to have only a minimal effect on the resultant Young\u27s modulus. Predictive equations for Poisson\u27s ratio are second-order and dependent upon the void arrangement. The effect of changes in loading direction on moduli indicate that the three microstructures evaluated in this study are anisotropic, with anisotropy increasing with both ψ and volume porosity. The predictive equations developed in this study were validated with AM fabrication and testing of prototypical Ti6Al4V spinal rods. Constructs of a rhombohedral (FCC) pore arrangement with 30% porosity showed an effective reduction of ~ 50% in Young\u27s modulus. Predicted values for flexural rigidity fell within 95% confidence intervals for the tested porous Ti6Al4V constructs, confirming a design methodology with the potential of reducing the flexural rigidity, and resulting bone resorption, of orthopaedic implants

Development of M5 Cladding Material Correlations in the TRANSURANUS Code: Revision 1

The technical report is based on an earlier research on material properties of the M5 structural material. Complementing this research with new M5 data found in open literature, a set of correlations has been developed for the implementation to the TRANSURANUS code. This includes thermal, mechanical, and chemical (corrosion) properties of M5. As an example, thermal capacity or burst stress correlations have been proposed using the available experimental data. The open literature provides a wide range of experimental data on M5, but for some quantities they are not complete enough to be suitable for the implementation to the TRANSURANUS code. A balanced consideration of similarity of M5 characteristics to those of Zircaloy-4 (Zry-4) or E110 have therefore led to the recommendation to use some of these data selectively also for M5. As such, creep anisotropy coefficients of E110 are recommended to be used also for M5.JRC.G.I.4-Nuclear Reactor Safety and Emergency Preparednes

Robustness and Randomness

Robustness problems of computational geometry algorithms is a topic that has been subject to intensive research efforts from both computer science and mathematics communities. Robustness problems are caused by the lack of precision in computations involving floating-point instead of real numbers. This paper reviews methods dealing with robustness and inaccuracy problems. It discussed approaches based on exact arithmetic, interval arithmetic and probabilistic methods. The paper investigates the possibility to use randomness at certain levels of reasoning to make geometric constructions more robust

Pseudospectral collocation method for viscoelastic guided wave problems in generally anisotropic media

In Non-Destructive Evaluation (NDE) applications guided waves are attractive to perform rapid inspections of long lengths and large areas. However, they are complicated, therefore it is important to have as much information and understanding about their physical properties as possible in order to design the most efficient and robust inspection process as well as to draw the correct conclusions from the measurement results. The main piece of information to gain insight into the guided wave's properties is dispersion curves which, for isotropic structures such as plates and cylinders, have been available for many years. There are many robust algorithms which are currently used to compute them: finite element simulations, partial wave based root finding routines (PWRF) and semi-analytical finite element simulations (SAFE). These methodologies have been generalized and also used to study and compute dispersion curves of more complicated anisotropic materials though the range of tractable cases was limited. Although robust, all these approaches present several challenges, mostly computational, such as missing modes (PWRF), the so called "large-fd" problem (PWRF), artificially increased stiffness (FE, SAFE) or improvement of dispersion curve tracing routines (FE, PWRF, SAFE). In addition, when studying complicated anisotropic materials with a low degree of symmetry or unusual axes configurations where propagation does not take place along any of the principal axes, PWRF routines are frequently unreliable and one must resort to specific SAFE simulations which also present their own challenges and, depending on the SAFE scheme used, can yield spurious modes which need to be carefully filtered. Recently, Pseudospectral Methods (Galerkin and Collocation schemes), were introduced in the field of elastic guided waves, providing a powerful, yet strikingly and conceptually simple alternative to the above algorithms by successfully finding the dispersion curves in isotropic structures and in some simple anisotropic problems. However, a systematic and general approach for accurately and robustly computing dispersion curves of guided waves in anisotropic media, up to the most general case of triclinic symmetry, has not yet been developed. The goal of the work presented in this thesis is to develop such a tool by means of the Pseudospectral Collocation Method (SCM) and to take advantage of its particular features to make it as robust as possible. Firstly, a PSCM scheme is developed for computing dispersion curves of guided waves in anisotropic elastic media by finding all the frequencies for a given value of the real wavenumber. The results are validated with the existing literature as well as with the results provided by the software DISPERSE developed in the NDT group at Imperial College London. Many of the most remarkable features of the PSCM (spectral accuracy, speed, and its failure to miss modes for instance) are already observed in this simple, yet important, class of problems in elastic media. Secondly, guided waves in viscoelastic anisotropic media are studied. In this case, modes present attenuation due to material damping which is reflected in the wavenumber being complex. In order to handle complex wavenumbers the PSCM schemes developed for elastic materials are appropriately extended by means of the Companion Matrix Method. It will be seen that, apart from lowly attenuated propagating modes, all the other highly attenuated modes are found, yielding the full three-dimensional spectrum of the problem under consideration. Moreover, when the PSCM schemes for viscoelastic media are used to compute the dispersion curves of guided waves in an elastic medium, all the remaining, imaginary as well as complex, roots of the elastic problem which were not computed by the simpler PSCM elastic schemes are found, providing the full three-dimensional picture of the dispersion curves. These PSCM schemes, as any other of the aforementioned approaches, only find pairs (\omega,k). If dispersion curves are to be plotted, those pairs must be linked correctly in order to plot the desired dispersion curves, which is non-trivial when crossings amongst modes occur. Motivated by this, an investigation of the parity and coupling properties of guided wave solutions is carried out in detail for all crystal classes. This investigation provides a robust alternative to conventional tracing routines and avoids the problem of mode crossings by exploiting the parity and coupling properties of the solutions. Finally, the most complicated problems involving embedded structures are investigated by including a Perfectly Matched Layer (PML) in the previously developed PSCM schemes for viscoelastic media. The dispersion curves for leaky and trapped modes in an isotropic elastic plate and in a similar cylinder immersed in an infinite ideal fluid are found, showing very good agreement with the results given by PWRF routines in a large range of frequencies. Last, but not least, an illustration of a two-dimensional PSCM scheme is presented to study a vibrating membrane. The results are compared with the available analytical solution showing again excellent agreement.Open Acces