Spline.design Alternative
Gaetan Horton
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To simulate increasingly complex physical phenomena and systems, tightly integrated design-through-analysis (DTA) tools are essential. In this dissertation, the complementary strengths of isogeometric analysis and T-splines are coupled and enhanced to create a seamless DTA framework. In all cases, the technology de- veloped meets the demands of both design and analysis. In isogeometric analysis, the smooth geometric basis is used as the basis for analysis. It has been demonstrated that smoothness offers important computational advantages over standard finite elements. T-splines are a superior alternative to NURBS, the current geometry standard in computer-aided design systems. T-splines can be locally refined and can represent complicated designs as a single watertight geometry. These properties make T-splines an ideal discretization technology for isogeometric analysis and, on a higher level, a foundation upon which unified DTA technologies can be built.
We characterize analysis-suitable T-splines and develop corresponding finite element technology, including the appropriate treatment of extraordinary points (i.e., unstructured meshing). Analysis-suitable T-splines form a practically useful subset of T-splines. They maintain the design flexibility of T-splines, including an efficient and highly localized refinement capability, while preserving the important analysis-suitable mathematical properties of the NURBS basis.
We identify Bzier extraction as a unifying paradigm underlying all isogeometric element technology. Bzier extraction provides a finite element representation of NURBS or T-splines, and facilitates the incorporation of T-splines into existing finite element programs. Only the shape function subroutine needs to be modified. Additionally, Bzier extraction is automatic and can be applied to any T-spline regardless of topological complexity or polynomial degree. In particular, it represents an elegant treatment of T-junctions, referred to as "hanging nodes" in finite element analysis
Finally, we explore the behavior of T-splines in finite element analysis. It is demonstrated that T-splines possess similar convergence properties to NURBS with far fewer degrees of freedom. We develop an adaptive isogeometric analysis framework which couples analysis-suitable T-splines, local refinement, and Bzier extraction and apply it to the modeling of damage and fracture processes. These examples demonstrate the feasibility of applying T-spline element technology to very large problems in two and three dimensions and parallel implementations.
I am porting a script written in R over to Python. In R I am using smooth.spline and in Python I am using SciPy UnivariateSpline. They don't produce the same results (even though they are both based on a cubic spline method). Is there a way, or an alternative to UnivariateSpline, to make the Python spline return the same spline as R?
smooth.spline in R is a "smoothing spline", which is an overparametrized natural spline (knots at every data point, cubic spline in the interior, linear extrapolation), with penalized least squares used to choose the parameters. You can read the help page for the details of how the penalty is computed.
These are completely different algorithms, and I wouldn't expect them to give equal results.I don't know if there's an R package that uses the same adaptive choice of knots as Python does. This answer: claims to reference a natural smoothing spline implementation in Python, but I don't know if it matches R's implementation.
Note that this code is not fully compatible with Jupyter-notebooks for the latest versions of rpy2. You can fix this by using !pip install -Iv rpy2==3.4.2 as described in NotImplementedError: Conversion 'rpy2py' not defined for objects of type '' only after I run the code twice
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A novel design of stochastic numerical computing method is introduced for computational fluid dynamics problem governed with nonlinear thin film flow (TFF) system by exploiting the competency of polynomial splines for discretization and optimization with evolutionary computing aided with brilliance of local search. The TFF model of second grade fluid is represented with nonlinear second-order differential system. The aim of the present work is to exploit the cubic spline approach (CSA) to transform the differential equations for TFF model into an equivalent set of nonlinear equations. The approximation in mean squared error sense is introduced for the formulation of cost function for solving the nonlinear system of equations representing TFF model. The optimization of the decision variables of the cost function is carried out with global search efficacy of evolution by genetic algorithms (GAs) integrated with sequential quadratic programming (SQP) for speedy adjustments. The designed spline-evolutionary computing paradigm, CSA-GA-SQP, is evaluated for different scenarios of TFF model by variation of second grade and magnetic parameters, as well as variation in the length of splines. Results endorsed the worth of CSA-GA-SQP solver as an efficient alternative, reliable, stable, and accurate framework for the variants of nonlinear TFF systems on the basis of multiple autonomous executions. The design computing spline paradigm CSA-GA-SQP is a promising alternative numerical solver to be implemented for the solution of stiff nonlinear systems representing the complex scenarios of computational fluid dynamics problems.
Spline is a Web-based, Realtime, Collaborative 3D design software which can create, model, sculpt and animate 3D objects.It can create organic shapes from scratch and users can collaborate on projects seamlessly.It also features applying physics-based interactions to the model created. View or download Spline via spline.design.
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There is no such thing as a strength of a wrench by itself. Its strength depends on the type and strength of the fastener on which it is used and the clearance between the wrench on the fastener. No more torque will be transmitted than the strength of its weakest member, be it the wrench or the fastener. It is much better if the wrench is stronger than the fastener.
Traditionally, fasteners have been designed before wrenches, or to use with existing wrenches. Head size, shape and tensile strength were all factors beyond the control of the wrench manufacturer. Early wrenches were designed with sufficient outside diameter to provide ample torsional strength to turn existing fasteners. Product manufacturers noted wrench dimensions and provided enough space for wrenches but no excess because that often-increased product cost.
The second big change has been the increased strength of many fasteners. Fasteners have gone from 60,000 psi tensile strength to over 180,000 psi, a tripled increase. In addition, the use of the small 12-point fastener heads requires smaller wrenches to have twice their customary strength required to turn 6-point fasteners. The combined result is a sixfold increase in required strength.
Double hex, or 12-point fasteners, have been used for increased torque or smaller fastener heads. These fasteners may be turned with either standard or premium quality 12-point wrenches. While 12-point wrenches do a fairly good job, they are not completely reliable because of the substantial variation that is found in removal torque, depending both on the tightness to which the fastener was originally tightened and its exposure to corrosion and overload.
Although there are limitations with 12-point fasteners, we should realize that there is alternative in spline wrenches and spline fasteners which would deliver twice as much torque while requiring no additional clearance space because of the inherent efficiency of the spline design. See illustration of shapes.
The combination of spline wrenches and spline fasteners can deliver twice the torque of 12-point wrenches on double hex fasteners and still more torque than hex wrenches on hex fasteners. This is done without any increase in the outside diameter of the wrench. This allows the use of higher strength bolts, and it allows these fasteners to be tightened to higher loads. Even more important, it allows these fasteners to be removed with less difficulty. Like many new ideas, they first used in aerospace, where weight and time savings justify their premium cost. Also, splines are a good replacement for hollow head cap screws because they can work in the same space, and a new design is easier to clean than the hole of a socket head cap screw.
If a trades person has a set of spline wrenches, they will find that they will turn single and double hex 12-point fasteners with corners too severely damaged to be turned with other wrenches. Also, they are superior for double hex fasteners because they fit in the square space. Some users find that it pays to replace double hex headed fasteners with spline fasteners to make them easier to remove in the future.
Spline fasteners help tighten high tensile strength spline fasteners which are used in high torque load applications. The spline profile allows a higher torque to be applied before rounding of the fastener corners occurs. Applied loads are moved away from the fastener corners which allow the spline wrench to turn fasteners that are already worn or rounded.
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