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All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

Spatial adaptation procedures on tetrahedral meshes for unsteady aerodynamic flow calculations

Spatial adaptation procedures for the accurate and efficient solution of steady and unsteady inviscid flow problems are described. The adaptation procedures were developed and implemented within a three-dimensional, unstructured-grid, upwind-type Euler code. These procedures involve mesh enrichment and mesh coarsening to either add points in high gradient regions of the flow or remove points where they are not needed, respectively, to produce solutions of high spatial accuracy at minimal computational cost. The paper gives a detailed description of the enrichment and coarsening procedures and presents comparisons with experimental data for an ONERA M6 wing and an exact solution for a shock-tube problem to provide an assessment of the accuracy and efficiency of the capability. Steady and unsteady results, obtained using spatial adaptation procedures, are shown to be of high spatial accuracy, primarily in that discontinuities such as shock waves are captured very sharply.

Rausch, Russ D.↗
NASA NTRS1992

Automated assembly of a tetrahedral truss structure using machine vision

The Automated Structures Assembly Laboratory is a unique facility at NASA Langley Research Center used to investigate the robotic assembly of truss structures. Two special-purpose end-effectors have been used to assemble 102 truss members and 12 panels into an 8-meter diameter structure. One end-effector is dedicated to truss member insertion, while a second end-effector is used to install panels. Until recently, the robot motions required to construct the structure were developed iteratively using the facility hardware. Recent work at Langley has resulted in a compact machine vision system capable of providing position information relative to targets on the structure. Use of the vision system to guide the robot from an approach point 10 to 18 inches from the structure, offsetting model inaccuracies, permits robot motion based on calculated points as a first step toward use of preplanned paths from an automated path planner. This paper presents recent work at Langley highlighting the application of the machine vision system during truss member insertion.

Doggett, William R.↗
NASA NTRS1993

Tetrahedral lander

An apparatus and method is disclosed for decelerating and absorbing impact of a re-entry vehicle suitable for payloads that are relatively light as well as payloads weighing several tons or more. The apparatus includes four inflatable legs displaced equidistantly from each other around a capsule or housing which contains a payload. The legs are inflated at a designated altitude after entering earth's atmosphere to slow the descent of the re-entry vehicle. Connected between each of the four legs are drag inducing surfaces that deploy as the legs inflate. The drag inducing surfaces are triangularly shaped with one such surface being connected between each pair of legs for a total of six drag inducing surfaces. The legs have drag inducing outer surfaces which act to slow the descent of the re-entry vehicle.

Roberts, Michael L.↗
NASA NTRS1993

Dynamic mesh adaption for triangular and tetrahedral grids

The following topics are discussed: requirements for dynamic mesh adaption; linked-list data structure; edge-based data structure; adaptive-grid data structure; three types of element subdivision; mesh refinement; mesh coarsening; additional constraints for coarsening; anisotropic error indicator for edges; unstructured-grid Euler solver; inviscid 3-D wing; and mesh quality for solution-adaptive grids. The discussion is presented in viewgraph form.

Rupak Biswas↗
NASA NTRS1995

Adaptive hybrid prismatic-tetrahedral grids for viscous flows

The paper presents generation of adaptive hybrid prismatic/tetrahedral grids for complex 3-D geometries including multi-body domains. The prisms cover the region close to each body's surface, while tetrahedra are created elsewhere. Two developments are presented for hybrid grid generation around complex 3-D geometries. The first is a new octree/advancing front type of method for generation of the tetrahedra of the hybrid mesh. The main feature of the present advancing front tetrahedra generator that is different from previous such methods is that it does not require the creation of a background mesh by the user for the determination of the grid-spacing and stretching parameters. These are determined via an automatically generated octree. The second development is an Automatic Receding Method (ARM) for treating the narrow gaps in between different bodies in a multiply-connected domain. This method is applied to a two-element wing case. A hybrid grid adaptation scheme that employs both h-refinement and redistribution strategies is developed to provide optimum meshes for viscous flow computations. Grid refinement is a dual adaptation scheme that couples division of tetrahedra, as well as 2-D directional division of prisms.

Kallinderis, Yannis↗
NASA NTRS1999

Parallel Tetrahedral Mesh Adaptation with Dynamic Load Balancing

The ability to dynamically adapt an unstructured grid is a powerful tool for efficiently solving computational problems with evolving physical features. In this paper, we report on our experience parallelizing an edge-based adaptation scheme, called 3D_TAG. using message passing. Results show excellent speedup when a realistic helicopter rotor mesh is randomly refined. However. performance deteriorates when the mesh is refined using a solution-based error indicator since mesh adaptation for practical problems occurs in a localized region., creating a severe load imbalance. To address this problem, we have developed PLUM, a global dynamic load balancing framework for adaptive numerical computations. Even though PLUM primarily balances processor workloads for the solution phase, it reduces the load imbalance problem within mesh adaptation by repartitioning the mesh after targeting edges for refinement but before the actual subdivision. This dramatically improves the performance of parallel 3D_TAG since refinement occurs in a more load balanced fashion. We also present optimal and heuristic algorithms that, when applied to the default mapping of a parallel repartitioner, significantly reduce the data redistribution overhead. Finally, portability is examined by comparing performance on three state-of-the-art parallel machines.

Oliker, Leonid↗
NASA NTRS2000

A Solution Adaptive Technique Using Tetrahedral Unstructured Grids

An adaptive unstructured grid refinement technique has been developed and successfully applied to several three dimensional inviscid flow test cases. The method is based on a combination of surface mesh subdivision and local remeshing of the volume grid Simple functions of flow quantities are employed to detect dominant features of the flowfield The method is designed for modular coupling with various error/feature analyzers and flow solvers. Several steady-state, inviscid flow test cases are presented to demonstrate the applicability of the method for solving practical three-dimensional problems. In all cases, accurate solutions featuring complex, nonlinear flow phenomena such as shock waves and vortices have been generated automatically and efficiently.

Pirzadeh, Shahyar Z.↗
NASA NTRS2004

Optimal Reconfiguration of Tetrahedral Formations

The problem of minimum-fuel formation reconfiguration for the Magnetospheric Multi-Scale (MMS) mission is studied. This reconfiguration trajectory optimization problem can be posed as a nonlinear optimal control problem. In this research, this optimal control problem is solved using a spectral collocation method called the Gauss pseudospectral method. The objective of this research is to provide highly accurate minimum-fuel solutions to the MMS formation reconfiguration problem and to gain insight into the underlying structure of fuel-optimal trajectories.

Huntington, Geoffrey↗
NASA NTRS2007

Agent Based Intelligence in a Tetrahedral Rover

A tetrahedron is a 4-node 6-strut pyramid structure which is being used by the NASA - Goddard Space Flight Center as the basic building block for a new approach to robotic motion. The struts are extendable; it is by the sequence of activities: strut-extension, changing the center of gravity and falling that the tetrahedron "moves". Currently, strut-extension is handled by human remote control. There is an effort underway to make the movement of the tetrahedron autonomous, driven by an attempt to achieve a goal. The approach being taken is to associate an intelligent agent with each node. Thus, the autonomous tetrahedron is realized as a constrained multi-agent system, where the constraints arise from the fact that between any two agents there is an extendible strut. The hypothesis of this work is that, by proper composition of such automated tetrahedra, robotic structures of various levels of complexity can be developed which will support more complex dynamic motions. This is the basis of the new approach to robotic motion which is under investigation. A Java-based simulator for the single tetrahedron, realized as a constrained multi-agent system, has been developed and evaluated. This paper reports on this project and presents a discussion of the structure and dynamics of the simulator.

Phelps, Peter↗
NASA NTRS2011

Adaptive refinement tools for tetrahedral unstructured grids

An exemplary embodiment providing one or more improvements includes software which is robust, efficient, and has a very fast run time for user directed grid enrichment and flow solution adaptive grid refinement. All user selectable options (e.g., the choice of functions, the choice of thresholds, etc.), other than a pre-marked cell list, can be entered on the command line. The ease of application is an asset for flow physics research and preliminary design CFD analysis where fast grid modification is often needed to deal with unanticipated development of flow details.

Pao, S. Paul↗
NASA NTRS2013

Recent Development in the CESE Method for the Solution of the Navier-Stokes Equations Using Unstructured Triangular or Tetrahedral Meshes With High Aspect Ratio

In the multidimensional CESE development, triangles and tetrahedra turn out to be the most natural building blocks for 2D and 3D spatial meshes. As such the CESE method is compatible with the simplest unstructured meshes and thus can be easily applied to solve problems with complex geometries. However, because the method uses space-time staggered stencils, solution decoupling may become a real nuisance in applications involving unstructured meshes. In this paper we will describe a simple and general remedy which, according to numerical experiments, has removed any possibility of solution decoupling. Moreover, in a real-world viscous flow simulation near a solid wall, one often encounters a case where a boundary with high curvature or sharp corner is surrounded by triangular/tetrahedral meshes of extremely high aspect ratio (up to 106). For such an extreme case, the spatial projection of a space-time compounded conservation element constructed using the original CESE design may become highly concave and thus its centroid (referred to as a spatial solution point) may lie far outside of the spatial projection. It could even be embedded beyond a solid wall boundary and causes serious numerical difficulties. In this paper we will also present a new procedure for constructing conservation elements and solution elements which effectively overcomes the difficulties associated with the original design. Another difficulty issue which was addressed more recently is the wellknown fact that accuracy of gradient computations involving triangular/tetrahedral grids deteriorates rapidly as the aspect ratio of grid cells increases. The root cause of this difficulty was clearly identified and several remedies to overcome it were found through a rigorous mathematical analysis. However, because of the length of the current paper and the complexity of mathematics involved, this new work will be presented in another paper.

Chang, Sin-Chung↗
NASA NTRSDate not supplied

Towards a Third-Order Accurate, Second-Derivative-Free, Shock-Capturing Finite-Volume Method for Hypersonic Flows on Tetrahedral Grids

In this paper, we report progress in the development of a third-order accurate, second-derivative-free, shock-capturing finite-volume solver for three-dimensional unstructured grids. The method is economical in the sense that the computation and storage of second derivatives are not required for third-order accuracy. It is based on point-valued numerical solutions stored at cells, gradients computed and stored at nodes, and an efficient projected-derivative formula that eliminates the need for second derivatives in a quadratic solution interpolation. The projected-derivative formula is also used to eliminate second derivatives from a high-order flux quadrature formula, so that it can be implemented conveniently in the form of a numerical flux at a face center plus a correction term. Similarly, a high-order source quadrature formula can also be implemented in the form of a cell-center point evaluation plus a similar correction term. These features make it relatively straightforward to extend an existing second-order finite-volume code to third-order. This paper reports progress of implementing the method in the NASA VULCAN-CFD code and discusses the implementation of a high-order accurate limiter for shock capturing.

Weighted Least-Squares↗
NASA NTRSDate not supplied

Towards Robust and Accurate Implicit Gradient Methods for Second- and Third-Order Nodal-Gradient Cell-Centered Finite-Volume Discretizations on Tetrahedral Grids

In this paper, we introduce implicit gradient methods as alternatives to conventional least-squares gradient methods for second- and third-order nodal-gradient cell-centered finite-volume discretizations, where solutions are stored at cells but gradients are stored at nodes. Because of the unique configuration of solutions and gradients, implicit gradient systems developed for the node-centered edge-based discretization method can be directly applied once the numerical solutions are interpolated from cells to nodes with sufficient accuracy. The resulting defect-correction solver can be loosely coupled with a flow-equation solver, and at convergence, solutions and gradients that satisfy the corresponding residual equations are obtained. Each iteration is relatively cheap compared with least-squares methods involving hundreds of neighbors. Numerical results are presented for accuracy verification studies and some simple but realistic flow problems.

Computational Fluid Dynamics↗
DOE OSTI2020

Materials Data on La20Al3Si9O52 by Materials Project

La20Al3Si9O52 crystallizes in the triclinic P1 space group. The structure is three-dimensional. there are twenty inequivalent La sites. In the first La site, La is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of La–O bond distances ranging from 2.45–2.93 Å. In the second La site, La is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of La–O bond distances ranging from 2.41–2.90 Å. In the third La site, La is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of La–O bond distances ranging from 2.49–2.92 Å. In the fourth La site, La is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of La–O bond distances ranging from 2.45–2.92 Å. In the fifth La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.32–2.82 Å. In the sixth La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.35–2.91 Å. In the seventh La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.35–2.85 Å. In the eighth La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.34–2.93 Å. In the ninth La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.34–2.92 Å. In the tenth La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.33–2.93 Å. In the eleventh La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.34–3.00 Å. In the twelfth La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.35–2.85 Å. In the thirteenth La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.31–2.90 Å. In the fourteenth La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.35–2.91 Å. In the fifteenth La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.35–2.90 Å. In the sixteenth La site, La is bonded in a 7-coordinate geometry to seven O atoms. There are a spread of La–O bond distances ranging from 2.33–2.84 Å. In the seventeenth La site, La is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of La–O bond distances ranging from 2.42–2.99 Å. In the eighteenth La site, La is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of La–O bond distances ranging from 2.39–2.98 Å. In the nineteenth La site, La is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of La–O bond distances ranging from 2.46–3.00 Å. In the twentieth La site, La is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of La–O bond distances ranging from 2.42–2.99 Å. There are three inequivalent Al sites. In the first Al site, Al is bonded in a tetrahedral geometry to four O atoms. There are a spread of Al–O bond distances ranging from 1.76–1.78 Å. In the second Al site, Al is bonded in a tetrahedral geometry to four O atoms. There are a spread of Al–O bond distances ranging from 1.76–1.78 Å. In the third Al site, Al is bonded in a tetrahedral geometry to four O atoms. There are a spread of Al–O bond distances ranging from 1.76–1.78 Å. There are nine inequivalent Si sites. In the first Si site, Si is bonded in a tetrahedral geometry to four O atoms. There are a spread of Si–O bond distances ranging from 1.64–1.66 Å. In the second Si site, Si is bonded in a tetrahedral geometry to four O atoms. There is one shorter (1.64 Å) and three longer (1.66 Å) Si–O bond length. In the third Si site, Si is bonded in a tetrahedral geometry to four O atoms. There are a spread of Si–O bond distances ranging from 1.64–1.66 Å. In the fourth Si site, Si is bonded in a tetrahedral geometry to four O atoms. There are a spread of Si–O bond distances ranging from 1.64–1.66 Å. In the fifth Si site, Si is bonded in a tetrahedral geometry to four O atoms. There is one shorter (1.64 Å) and three longer (1.66 Å) Si–O bond length. In the sixth Si site, Si is bonded in a tetrahedral geometry to four O atoms. There is one shorter (1.64 Å) and three longer (1.66 Å) Si–O bond length. In the seventh Si site, Si is bonded in a tetrahedral geometry to four O atoms. There is one shorter (1.64 Å) and three longer (1.65 Å) Si–O bond length. In the eighth Si site, Si is bonded in a tetrahedral geometry to four O atoms. There are a spread of Si–O bond distances ranging from 1.64–1.66 Å. In the ninth Si site, Si is bonded in a tetrahedral geometry to four O atoms. There are a spread of Si–O bond distances ranging from 1.64–1.66 Å. There are fifty-two inequivalent O sites. In the first O site, O is bonded in a 1-coordinate geometry to three La and one Al atom. In the second O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the third O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the fourth O site, O is bonded in a 1-coordinate geometry to three La and one Al atom. In the fifth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the sixth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the seventh O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the eighth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the ninth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the tenth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the eleventh O site, O is bonded in a 1-coordinate geometry to three La and one Al atom. In the twelfth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the thirteenth O site, O is bonded in a distorted tetrahedral geometry to three La and one Al atom. In the fourteenth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the fifteenth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the sixteenth O site, O is bonded in a distorted tetrahedral geometry to three La and one Al atom. In the seventeenth O site, O is bonded in a distorted single-bond geometry to three La and one Si atom. In the eighteenth O site, O is bonded in a distorted single-bond geometry to three La and one Si atom. In the nineteenth O site, O is bonded in a distorted single-bond geometry to three La and one Si atom. In the twentieth O site, O is bonded in a distorted single-bond geometry to three La and one Si atom. In the twenty-first O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the twenty-second O site, O is bonded in a distorted single-bond geometry to three La and one Si atom. In the twenty-third O site, O is bonded in a distorted tetrahedral geometry to three La and one Al atom. In the twenty-fourth O site, O is bonded in a distorted single-bond geometry to three La and one Si atom. In the twenty-fifth O site, O is bonded in a trigonal planar geometry to three La atoms. In the twenty-sixth O site, O is bonded in a trigonal planar geometry to three La atoms. In the twenty-seventh O site, O is bonded in a trigonal planar geometry to three La atoms. In the twenty-eighth O site, O is bonded in a trigonal planar geometry to three La atoms. In the twenty-ninth O site, O is bonded in a 1-coordinate geometry to three La and one Al atom. In the thirtieth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the thirty-first O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the thirty-second O site, O is bonded in a 1-coordinate geometry to three La and one Al atom. In the thirty-third O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the thirty-fourth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the thirty-fifth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the thirty-sixth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the thirty-seventh O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the thirty-eighth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the thirty-ninth O site, O is bonded in a 1-coordinate geometry to three La and one Al atom. In the fortieth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the forty-first O site, O is bonded in a 1-coordinate geometry to three La and one Al atom. In the forty-second O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the forty-third O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the forty-fourth O site, O is bonded in a 1-coordinate geometry to three La and one Al atom. In the forty-fifth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the forty-sixth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the forty-seventh O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the forty-eighth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the forty-ninth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the fiftieth O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the fifty-first O site, O is bonded in a 1-coordinate geometry to three La and one Si atom. In the fifty-second O site, O is bonded in a 1-coordinate geometry to three La and one Al atom.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Materials Data on Sr20P12Cl(O16F)3 by Materials Project

Sr20P12Cl(O16F)3 crystallizes in the triclinic P1 space group. The structure is three-dimensional. there are twenty inequivalent Sr sites. In the first Sr site, Sr is bonded in a 7-coordinate geometry to six O and one Cl atom. There are a spread of Sr–O bond distances ranging from 2.41–2.77 Å. The Sr–Cl bond length is 2.78 Å. In the second Sr site, Sr is bonded in a 7-coordinate geometry to six O and one F atom. There are a spread of Sr–O bond distances ranging from 2.51–2.86 Å. The Sr–F bond length is 2.44 Å. In the third Sr site, Sr is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of Sr–O bond distances ranging from 2.61–2.95 Å. In the fourth Sr site, Sr is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of Sr–O bond distances ranging from 2.60–2.96 Å. In the fifth Sr site, Sr is bonded in a 7-coordinate geometry to six O and one F atom. There are a spread of Sr–O bond distances ranging from 2.52–2.81 Å. The Sr–F bond length is 2.45 Å. In the sixth Sr site, Sr is bonded in a 7-coordinate geometry to six O and one F atom. There are a spread of Sr–O bond distances ranging from 2.54–2.84 Å. The Sr–F bond length is 2.45 Å. In the seventh Sr site, Sr is bonded in a 7-coordinate geometry to six O and one Cl atom. There are a spread of Sr–O bond distances ranging from 2.44–2.80 Å. The Sr–Cl bond length is 2.78 Å. In the eighth Sr site, Sr is bonded in a 7-coordinate geometry to six O and one Cl atom. There are a spread of Sr–O bond distances ranging from 2.44–2.76 Å. The Sr–Cl bond length is 2.79 Å. In the ninth Sr site, Sr is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of Sr–O bond distances ranging from 2.59–3.05 Å. In the tenth Sr site, Sr is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of Sr–O bond distances ranging from 2.60–2.96 Å. In the eleventh Sr site, Sr is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of Sr–O bond distances ranging from 2.58–3.03 Å. In the twelfth Sr site, Sr is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of Sr–O bond distances ranging from 2.60–2.96 Å. In the thirteenth Sr site, Sr is bonded in a 7-coordinate geometry to six O and one F atom. There are a spread of Sr–O bond distances ranging from 2.55–2.83 Å. The Sr–F bond length is 2.44 Å. In the fourteenth Sr site, Sr is bonded in a 7-coordinate geometry to six O and one F atom. There are a spread of Sr–O bond distances ranging from 2.55–2.83 Å. The Sr–F bond length is 2.45 Å. In the fifteenth Sr site, Sr is bonded in a 7-coordinate geometry to six O and one F atom. There are a spread of Sr–O bond distances ranging from 2.53–2.81 Å. The Sr–F bond length is 2.45 Å. In the sixteenth Sr site, Sr is bonded in a 7-coordinate geometry to six O and one F atom. There are a spread of Sr–O bond distances ranging from 2.51–2.80 Å. The Sr–F bond length is 2.45 Å. In the seventeenth Sr site, Sr is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of Sr–O bond distances ranging from 2.56–3.05 Å. In the eighteenth Sr site, Sr is bonded in a 9-coordinate geometry to nine O atoms. There are a spread of Sr–O bond distances ranging from 2.59–3.02 Å. In the nineteenth Sr site, Sr is bonded in a 7-coordinate geometry to six O and one F atom. There are a spread of Sr–O bond distances ranging from 2.52–2.81 Å. The Sr–F bond length is 2.45 Å. In the twentieth Sr site, Sr is bonded in a 7-coordinate geometry to six O and one F atom. There are a spread of Sr–O bond distances ranging from 2.54–2.84 Å. The Sr–F bond length is 2.44 Å. There are twelve inequivalent P sites. In the first P site, P is bonded in a tetrahedral geometry to four O atoms. All P–O bond lengths are 1.56 Å. In the second P site, P is bonded in a tetrahedral geometry to four O atoms. All P–O bond lengths are 1.56 Å. In the third P site, P is bonded in a tetrahedral geometry to four O atoms. There is two shorter (1.55 Å) and two longer (1.56 Å) P–O bond length. In the fourth P site, P is bonded in a tetrahedral geometry to four O atoms. All P–O bond lengths are 1.56 Å. In the fifth P site, P is bonded in a tetrahedral geometry to four O atoms. All P–O bond lengths are 1.56 Å. In the sixth P site, P is bonded in a tetrahedral geometry to four O atoms. All P–O bond lengths are 1.56 Å. In the seventh P site, P is bonded in a tetrahedral geometry to four O atoms. All P–O bond lengths are 1.56 Å. In the eighth P site, P is bonded in a tetrahedral geometry to four O atoms. All P–O bond lengths are 1.56 Å. In the ninth P site, P is bonded in a tetrahedral geometry to four O atoms. All P–O bond lengths are 1.56 Å. In the tenth P site, P is bonded in a tetrahedral geometry to four O atoms. All P–O bond lengths are 1.56 Å. In the eleventh P site, P is bonded in a tetrahedral geometry to four O atoms. There is two shorter (1.55 Å) and two longer (1.56 Å) P–O bond length. In the twelfth P site, P is bonded in a tetrahedral geometry to four O atoms. There is two shorter (1.55 Å) and two longer (1.56 Å) P–O bond length. There are forty-eight inequivalent O sites. In the first O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the second O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the third O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the fourth O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the fifth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the sixth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the seventh O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the eighth O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the ninth O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the tenth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the eleventh O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the twelfth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the thirteenth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the fourteenth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the fifteenth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the sixteenth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the seventeenth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the eighteenth O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the nineteenth O site, O is bonded in a distorted tetrahedral geometry to three Sr and one P atom. In the twentieth O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the twenty-first O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the twenty-second O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the twenty-third O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the twenty-fourth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the twenty-fifth O site, O is bonded in a 4-coordinate geometry to three Sr and one P atom. In the twenty-sixth O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the twenty-seventh O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the twenty-eighth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the twenty-ninth O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the thirtieth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the thirty-first O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the thirty-second O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the thirty-third O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the thirty-fourth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the thirty-fifth O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the thirty-sixth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the thirty-seventh O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the thirty-eighth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the thirty-ninth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the fortieth O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the forty-first O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the forty-second O site, O is bonded in a 4-coordinate geometry to three Sr and one P atom. In the forty-third O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the forty-fourth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. In the forty-fifth O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the forty-sixth O site, O is bonded in a distorted tetrahedral geometry to three Sr and one P atom. In the forty-seventh O site, O is bonded in a distorted single-bond geometry to three Sr and one P atom. In the forty-eighth O site, O is bonded in a 1-coordinate geometry to three Sr and one P atom. Cl is bonded in a trigonal non-coplanar geometry to three Sr atoms. There are three inequivalent F sites. In the first F site, F is bonded in a trigonal planar geometry to three Sr atoms. In the second F site, F is bonded in a trigonal planar geometry to three Sr atoms. In the third F site, F is bonded in a trigonal planar geometry to three Sr atoms.

36 MATERIALS SCIENCE↗