Edge and Face Meshing
|
|
- Ezra Richards
- 5 years ago
- Views:
Transcription
1 dge and Face Meshing 5-1
2 Meshing - General To reduce overall mesh size, confine smaller cells to areas where they are needed Locations of large flow field gradients. Locations of geometric details you wish to resolve. Controlling cell size distribution dges, faces, and volumes can be meshed directly. A uniform mesh is generated unless pre-meshing or size functions are used. Pre-meshing dge meshes can be graded (varying interval size on edge) A graded edge mesh can be used to control the cell size distribution of a face mesh. Controlling distribution of cell size on face mesh also controls the cell size distribution of the volume mesh. ize functions and boundary layers Allow direct control of cell size distribution on edges, faces and volumes directly for automatic meshing. 5-2
3 dge Meshing dge mesh distribution is controlled through the spacing and grading parameters on the Mesh dges form. Picking Temporary graphics Links, Directions Grading/pacing pecial characteristics Apply and Defaults Invert and Reverse Options 5-3
4 dge Meshing ense ense is used to show direction of grading very picked edge will show its sense direction using an arrow The sense can be reversed by a shift+middle-click on the last edge picked (this is in addition to the next functionality) or by clicking the Reverse button dge mesh preview When you pick an edge, the edge mesh is displayed using white nodes. This is a temporary mesh that has not been applied to the edge. Displayed edge mesh is based on current grading and spacing parameters If you modify the grading or spacing, the temporary mesh will be updated immediately. Meshing the edge The edge mesh is generated by clicking the Apply button. The nodes will then be displayed in blue. 5-4
5 Grading Controls mesh density distribution along an edge. Grading can produce single-sided or double-sided mesh Doubled-sided mesh can be symmetric or asymmetric. ymmetric schemes produce symmetric mesh about edge center. Asymmetric schemes can produce asymmetric mesh about edge center. ingle-sided grading: Uses a multiplicative constant, R, to describe the ratio of the length of two adjacent mesh elements: ingle-sided grading ymmetric grading Asymmetric grading R can be a user-specified value (uccessive Ratio) or calculated by GAMBIT. GAMBIT also uses edge length and spacing information to determine R. 5-5
6 Double ided Grading Double-sided grading can be generated by activating the double sided option in the Mesh dge form. Asymmetric grading is possible when the double-sided option is used with: uccessive Ratio, First Length, Last Length, First-Last Ratio, and Last-First Ratio The mesh is symmetric if R 1 = R 2 The mesh is asymmetric if R 1 R 2. dge center is determined automatically. ome schemes implicitly generate double sided grading that is symmetric. 5-6
7 oft Links Picking and soft links Pick with links By enabling this option, soft-linked edges can be selected in a single pick Linked edges share the same information and can be picked in a single pick Modifying soft links At any time, you can Form links Break links Maintain links By default, GAMBIT will form links between unmeshed edges that are picked together By default, GAMBIT will maintain links between meshed edges that are picked together 5-7
8 pacing In all meshing forms, the following spacing functions can be specified: Interval Count (recommended for edge mesh only) xample ntering a value of 5 will create 5 intervals along the selected edge(s) (6 nodes, including end nodes) Interval ize (default setting) Requires input of distance between nodes. dge is meshed with average interval size if grading is used. xample: An edge length of 10 and a value of 2 creates 5 intervals on the edge hortest dge % Meshes the selected edge according to a percentage of the length of the shortest edge in the model. xample hortest edge in model has length of 1. ntering a value of 20 will create a mesh with interval size
9 First dge ettings Use First dge ettings option If enabled: First edge selected in pick list updates all entries in the form. This mode is useful to copy settings from one meshed edge to other edge(s). If disabled: Use this setting any time you pick two or more meshed edges where there is a difference in type or spacing. The local Apply button for that option will be turned off This allows you to maintain pre-existing grading and/or spacing settings for each edge. nforce a change in grading and/or spacing by enabling Apply button. 5-9
10 Meshing Options Mesh This option is useful in cases where you want to impose a scheme without prescribing the number of intervals The higher level meshing scheme will decide (and match) the intervals Remove old mesh Deletes old mesh When selected, option to also delete lower geometry mesh appears. Ignore size function Toggle to either obey or ignore size functions ize function takes precedence when this option is disabled. 5-10
11 Meshing Options xample 1 pecify interval size, no grading, apply without meshing 3 Generate face mesh. Face Mesh Generated Using Quad Pave cheme (Pave face meshing schemes require an even number of elements on edge meshes) 2 pecify grading only, apply without meshing Face Mesh Generated Using ubmap cheme 5-11
12 Face Meshing Mesh Faces form Upon picking a face: GAMBIT automatically chooses quad elements GAMBIT chooses the type based on the solver/face vertex types Available element/scheme type combinations Quadrilaterial: Map, ubmap, Tri-Primitive, Pave Triangular: Pave Quad/Tri (hybrid): Map, Pave, Wedge Quad-to-tri conversion utility. 5-12
13 Face Meshing - Quad xamples Quad: Map Quad: ubmap Quad: Tri-Primitive Quad: Pave 5-13
14 Face Meshing - Quad/Tri and Tri xamples Quad/Tri Map Quad/Tri Wedge Face must be split to generate more than one cell across Quad/Tri Pave Tri Pave Triangular cell Quad cells Triangular cell 5-14
15 Deleting Old Mesh xisting mesh must be removed before remeshing. Mesh can be deleted using delete mesh form. Lower topology mesh can also be deleted (default) Alternatively, existing mesh can be deleted by selecting the Remove Old Mesh option Remove old mesh alone will leave all lower topology mesh Remove old mesh + remove lower mesh will delete all lower topology mesh that is not shared with another entity Undo after any meshing operation also works. 5-15
16 Face Vertex Types All vertices that are connected to a face are assigned initial face vertex types based on the angle between the edges connected to the vertex. Vertices shared by multiple faces can have multiple types, depending on which face you are considering. φ R The combination of vertex types describes the topology of the face. Face vertex types are used automatically to determine all quad face meshing schemes (except quad pave and tri pave). 5-16
17 Face Vertex Types nd () o 0 < φ <120 zero internal grid lines o ide () o 120 < φ < 216 One internal grid line o Corner (C) o 216 < φ < 309 Two internal grid lines o C C C Reverse (R) Three internal grid lines o 309 < φ < 360 o R R 5-17
18 Modifying Face Vertex Types Face vertex types can be changed from their default settings: Automatically By enforcing certain meshing schemes in face and volume meshing. Can sometimes result in undesirable mesh. Manually By direct modification in the Face Vertex Type form. elect Face symbols appear in graphics window elect New Vertex Type elect Vertices to be affected Vertex Types can be applied to just Boundary Layers as option. A vertex can have multiple types; one for each associated face. For a given set of face vertex types, GAMBIT will choose which meshing scheme to use based on predefined formulae. 5-18
19 Formula for ubmap cheme A face can be made submappable by manually changing vertex types Consider which vertex should be changed to type (side) In the et Face Vertex Type form, change vertex type to by enforcing the submap scheme. In the Face Mesh form, change the scheme from default to submap and click Apply. GAMBIT will attempt to change the vertex types so that the scheme is honored. User has less control the resulting mesh may be undesirable! R ubmap ( 4 nd) + ide + [( 2 nd) + Reverse] Which vertex to change? R R 5-19
20 Map ( 4 nd) + ( n ide) Formula for Map cheme + Periodic Map n ide Project intervals can be specified for more control. 5-20
21 How to Make a Face Mappable nforce the Map scheme (most common method) In the Face Mesh form, change the scheme from default to Map and click Apply. GAMBIT will attempt to change the vertex types so that the chosen scheme is honored Map ( 4 nd) + ( n ide) C C Manually change the vertex types In et Face Vertex Type form, change vertices (default) to "ide. Open the Face Mesh form and pick the face. GAMBIT should automatically select the map scheme) Map ( 4 nd) + ( n ide) 5-21
22 Formula for ubmap chemes ( ) ( ) ( ) ( ) ubmap: 4 nd + m ide + n nd + Corner + p Reverse + 2nd (additional terms when interior loops exist) C C Periodic ubmap n ide + m nd + Corner where m > 2. (additional terms when interior loops exist) ( ) ( ) C C C C C C C C C C C 5-22
23 Formula for Tri-Primitive heme Tri-Primitive ( 3 nd) + ( n ide) To mesh a rectangular face with the tri-primitive scheme: Manually change one of the vertex types to "ide" in this example The Tri Primitive scheme can not be enforced 5-23
24 Meshing Faces with Hybrid Quad/Tri chemes Quad/Tri: Tri-Map 2 Triangle The face vertex types must be changed manually to Trielement (T) The Tri-Map scheme must be selected. T T T Quad/Tri: Pave All vertex types are ignored except Trielement (T) and Notrielement (N) Trielement (T) will force a triangular element. Notrielement (N) will avoid a triangular element. C Quad/Tri: Wedge Used for creating cylindrical/polar type meshes The Vertex marked (T) is where rectangular elements are collapsed into triangles N 5-24 T
25 Hard Links Mesh linked entities have identical mesh Created for periodic boundary conditions Applicable to edge, face, and volume entities Best to use soft links for edge meshing To link volume meshes, all faces must be hard linked first. Hard links for faces elect faces and reference vertices The sense of each edge appears. Reverse orientation on by default Periodic option should be used for periodic boundary conditions, which creates a matched mesh even if the edges are split differently. Meshing one of the faces either before or after hard linking will generate an identical mesh on the linked face. Multiple pairs of hard links can be created. 5-25
26 Mesh moothing moothing can increase mesh quality beyond that of the default meshing algorithms Most noticable in complicated geometry. May have little or no effect in simple geometries. Mesh smoothing algorithms adjust interior node locations to obtain marginal improvement in mesh quality. Boundary meshes are not altered. The mesh at the boundary is not altered. Face and volume meshes are smoothed using a default scheme. Different schemes can be selected and applied after meshing. Face mesh smoothing Length-weighted Laplacian: Uses the average edge length of the elements surrounding each node to adjust the nodes. Centroid Area: Adjust node locations to equalize areas of adjacent elements. Winslow (quad meshes only): Optimizes element shapes with respect to perpendicularity. Volume mesh smoothing Length-weighted Laplacian: same as for face mesh smoothing quipotential: Adjusts node locations to equalize the volumes of the mesh elements surrounding each node. 5-26
27 xamining the Mesh Display Type Plane/phere View mesh elements that fall in plane or sphere. Range View mesh elements within quality range. Histogram shows quality distribution. how worst element zooms the view to the worst element elect 2D/3D and element type elect Quality Type Display Mode Change cell display attributes how Worst lement Automatically zooms the display to the worst element (based on current settings). Update button Will update values reported in the panel when options are changed. 5-27
28 Assessing Mesh Quality GAMBIT has several methods for assessing mesh quality. Aspect Ratio Diagonal Ratio dge Ratio quiangle kew quiize kew MidAngle kew ize Change tretch Taper Volume The most important of these quality metrics are quiangle kew and ize Change. 5-28
29 Mesh Quality quiangle kew The most important mesh quality metric is quiangle kew (Q A ). QA = θ θ max 180 θe max e, θ e θ θ e min θ max θ max θ θ max min θ e = Largest angle in face or cell = mallest angle in face or cell = Angle for equiangular face or cell θ min θ min Range of quiangle kew values 0 (best) 1 (worst) Q A = 0 describes a perfectly orthogonal element Q A = 1 describes a degenerate element θ e Quad/Hex o = 90 θ e θ e Tri/Tet = 60 θ e o 5-29
30 Mesh Quality ize Change Another important mesh quality metric is ize Change (QC). [ r, r, ] Q = max 2, C 1 K r n r i = Area or Volume of element i Area or Volume of neighbor element j V j=3 V j=2 Vi V j= 1 This metric applies only to 3D elements. V j=4 By definition, Q C > 0. 3D xample Q C,i = V j=1 /V i since j=1 has largest volume ratio Q C = 0 describes an element whose neighbor elements have exactly the same volume as the element of interest (i.e. uniform mesh). 5-30
31 triving for Quality A poor quality grid can cause inaccurate solutions and/or slow convergence. Minimize quiangle kew: Hex, Tri, Quad: kewness for all/most cells should be less than Tetrahedral: kewness for all/most cells should be less than 0.9. All elements: ize Change for cells in regions of interest should be less than 2 Minimize local variations in cell size, such as large jumps in size between adjacent cells. If xamine Mesh shows such violations: Determine the reason(s) for the violations Differences in spacing and grading on adjacent edges Geometry with small features or other defects Geometric complexity and size Mesh that grows too rapidly Delete mesh completely or partially. Clean and/or decompose geometry, premesh edges and faces or adjust meshing parameters Remesh the domain. 5-31
32 Appendix 5-32
33 Mesh Quality Aspect Ratio The Aspect Ratio metric (Q AR ) applies to tri, tet, quad, and hex elements and is defined differently for each element type. The definitions are as follows: Tri/Tet Inscribed circle Circumscribed circle Q AR = f is a scaling factor R and r are radii of circles (tri elements) or spheres (tet elements) that inscribe and circumscribe the mesh element. f = 1/2 for tri elements and f = 1/3 for tet elements. r f R r R 5-33 Quad/Hex Q AR max = min [ e1, e2, K, en ] [ e, e, K, e ] e i is the average length of edges in a coordinate system local to the element. N is the number of coordinate directions associated with the element d N = 2 for quad elements N = 3 for hex elements c a b 1 e e N a + c = 2 b + d = 2
34 Mesh Quality Diagonal Ratio The Diagonal Ratio metric (Q DR ) applies only to quad and hex elements and is defined as follows: Q DR max = min [ d1, d2, K, dn ] [ d, d, K, d ] The d i are the diagonals of the element. N is the total number of diagonals for a given element N = 2 for quad elements N = 4 for hex elements. 1 2 N d 3 d 1 d 2 d 1 d 4 d
35 Mesh Quality dge Ratio The dge Ratio quality metric (Q R ) is defined as follows: Q R max = min [ s1, s2, K, sn ] [ s, s, K, s ] The s i are the edge lengths of the element. N is the total number of edges for the element of interest. 1 2 N Tri N = 3 Quad N = 4 Tet N = 6 Pyramid N = 8 Wedge N = 9 Hex N =
36 Mesh Quality quiize kew The quiize kew metric (Q V ) applies only to quad and hex elements and is defined as follows: Q V = eq eq is the area (2D) or volume (3D) of the element of interest. eq is the maximum area (2D) or volume (3D) of an equilateral cell the circumscribing radius of which is identical to that of the mesh element. Actual element Area = 0 < Q V < 1 quilateral element Area = eq Q V =
37 Mesh Quality MidAngle kew The MidAngle kew (Q MA ) applies only to quadrilateral and hexahedral elements. Defined by the cosine of the minimum angle formed between the bisectors of the element edges (quad) or faces (hex). For quad elements: Q MA = cosθ For hex elements: Q MA = max [ cosθ,cosθ, θ ] 1 2 cos 3 θ Bisectors 5-37
38 Mesh Quality tretch The tretch quality metric (Q ) applies only to quadrilateral and hexahedral elements and is defined as follows: d i is the length of diagonal i s j is the length of the element edge j, Q [ s1, s2, K, sm ] [ d, d, K, d ] n and m are the total numbers of diagonals and edges, respectively. Quad elements: n = 2, m = 4, and K = 2; Hex elements: n = 4, m = 12, and K = 3. = 1 K min max 1 2 n By definition, 0 < Q < 1. Q = 0 describes an equilateral element Q = 1 describes a completely degenerate element. 5-38
39 Mesh Quality Taper The Taper quality metric (Q T ) applies only to quadrilateral and hexahedral mesh elements and is defined as follows: For any quadrilateral (or hexahedral) mesh element, it is possible to construct a parallelogram (or parallelepiped) such that the distance between any given corner of the parallelogram (or parallelepiped) and its nearest element corner node is a constant value. As a result, any vector, T, constructed from an element corner node to the nearest corner of the parallelogram (or parallelepiped) possesses a magnitude identical to that of all other such vectors. ach vector T can be resolved into components, T i, that are parallel to the bisectors of the mesh element. Quad elements: two components Hex elements: three components T 2 T T 1 Corner node lement edge Bisectors The Taper quality metric is defined as the normalized maximum of all such components for the element. By definition, 0 < Q T < 1. Q T = 0 describes an equilateral element Q T = 2 describes a degenerate element 5-39
40 Mesh Quality Volume and Warpage Volume The Volume quality metric (Q V ) applies only to 3D elements and represents quality in terms of element volume. Warpage The Warpage (Q W ) applies only to quad elements and is defined as follows: Z is the deviation from a best-fit plane that contains the element a and b are the lengths of the line segments that bisect the edges of the element. By definition, 0 < Q W < 1 Z Q W = min a, [ b] Q W = 0 describes an equilateral element Q W = 1 describes a degenerate element. lement edge Best-fit plane Bisectors 5-40
Reporting Mesh Statistics
Chapter 15. Reporting Mesh Statistics The quality of a mesh is determined more effectively by looking at various statistics, such as maximum skewness, rather than just performing a visual inspection. Unlike
More informationGEOMETRY MODELING & GRID GENERATION
GEOMETRY MODELING & GRID GENERATION Dr.D.Prakash Senior Assistant Professor School of Mechanical Engineering SASTRA University, Thanjavur OBJECTIVE The objectives of this discussion are to relate experiences
More informationIntroduction to Gambit 2.2. Training Notes
Introduction to Gambit 2.2 Training Notes Introduction to GAMBIT 1-1 What is GAMBIT? Geometry And Mesh Building Intelligent Toolkit A single, integrated preprocessor for CFD analysis: Geometry construction
More informationManipulating the Boundary Mesh
Chapter 7. Manipulating the Boundary Mesh The first step in producing an unstructured grid is to define the shape of the domain boundaries. Using a preprocessor (GAMBIT or a third-party CAD package) you
More informationLecture 7: Mesh Quality & Advanced Topics. Introduction to ANSYS Meshing Release ANSYS, Inc. February 12, 2015
Lecture 7: Mesh Quality & Advanced Topics 15.0 Release Introduction to ANSYS Meshing 1 2015 ANSYS, Inc. February 12, 2015 Overview In this lecture we will learn: Impact of the Mesh Quality on the Solution
More information3. MODELING A THREE-PIPE INTERSECTION (3-D)
3. MODELING A THREE-PIPE INTERSECTION (3-D) This tutorial employs primitives that is, predefined GAMBIT modeling components and procedures. There are two types of GAMBIT primitives: Geometry Mesh Geometry
More informationA NEW TYPE OF SIZE FUNCTION RESPECTING PREMESHED ENTITIES
A NEW TYPE OF SIZE FUNCTION RESPECTING PREMESHED ENTITIES Jin Zhu Fluent, Inc. 1007 Church Street, Evanston, IL, U.S.A. jz@fluent.com ABSTRACT This paper describes the creation of a new type of size function
More informationGeometry 10 and 11 Notes
Geometry 10 and 11 Notes Area and Volume Name Per Date 10.1 Area is the amount of space inside of a two dimensional object. When working with irregular shapes, we can find its area by breaking it up into
More informationHexahedral Meshing of Non-Linear Volumes Using Voronoi Faces and Edges
Hexahedral Meshing of Non-Linear Volumes Using Voronoi Faces and Edges Alla Sheffer and Michel Bercovier Institute of Computer Science, The Hebrew University, Jerusalem 91904, Israel. sheffa berco @cs.huji.ac.il.
More informationGeometry: Traditional Pathway
GEOMETRY: CONGRUENCE G.CO Prove geometric theorems. Focus on validity of underlying reasoning while using variety of ways of writing proofs. G.CO.11 Prove theorems about parallelograms. Theorems include:
More information15. SAILBOAT GEOMETRY
SAILBOAT GEOMETRY 15. SAILBOAT GEOMETRY In this tutorial you will import a STEP file that describes the geometry of a sailboat hull. You will split the hull along the symmetry plane, create a flow volume
More informationHPC Computer Aided CINECA
HPC Computer Aided Engineering @ CINECA Raffaele Ponzini Ph.D. CINECA SuperComputing Applications and Innovation Department SCAI 16-18 June 2014 Segrate (MI), Italy Outline Open-source CAD and Meshing
More informationGeometry Vocabulary. acute angle-an angle measuring less than 90 degrees
Geometry Vocabulary acute angle-an angle measuring less than 90 degrees angle-the turn or bend between two intersecting lines, line segments, rays, or planes angle bisector-an angle bisector is a ray that
More informationOverview of Unstructured Mesh Generation Methods
Overview of Unstructured Mesh Generation Methods Structured Meshes local mesh points and cells do not depend on their position but are defined by a general rule. Lead to very efficient algorithms and storage.
More information3. The sides of a rectangle are in ratio fo 3:5 and the rectangle s area is 135m2. Find the dimensions of the rectangle.
Geometry B Honors Chapter Practice Test 1. Find the area of a square whose diagonal is. 7. Find the area of the triangle. 60 o 12 2. Each rectangle garden below has an area of 0. 8. Find the area of the
More informationWest Windsor-Plainsboro Regional School District Basic Geometry Grades 9-12
West Windsor-Plainsboro Regional School District Basic Geometry Grades 9-12 Unit 1: Basics of Geometry Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale This unit
More informationMadison County Schools Suggested Geometry Pacing Guide,
Madison County Schools Suggested Geometry Pacing Guide, 2016 2017 Domain Abbreviation Congruence G-CO Similarity, Right Triangles, and Trigonometry G-SRT Modeling with Geometry *G-MG Geometric Measurement
More informationGeometry Geometry Grade Grade Grade
Grade Grade Grade 6.G.1 Find the area of right triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the
More information3 Identify shapes as two-dimensional (lying in a plane, flat ) or three-dimensional ( solid ).
Geometry Kindergarten Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). 1 Describe objects in the environment using names of shapes,
More informationGeometry Rules. Triangles:
Triangles: Geometry Rules 1. Types of Triangles: By Sides: Scalene - no congruent sides Isosceles - 2 congruent sides Equilateral - 3 congruent sides By Angles: Acute - all acute angles Right - one right
More informationYEC Geometry Scope and Sequence Pacing Guide
YEC Scope and Sequence Pacing Guide Quarter 1st 2nd 3rd 4th Units 1 2 3 4 5 6 7 8 G.CO.1 G.CO.2 G.CO.6 G.CO.9 G.CO.3 G.CO.7 G.CO.10 G.CO.4 G.CO.8 G.CO.11 Congruence G.CO.5 G.CO.12 G.CO.13 Similarity, Right
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY. 3 rd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY. 3 rd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More informationHexa Meshing. Defining Surface Parameters for the Mesh Defining Edge Parameters to Adjust the Mesh Checking mesh quality for determinants and angle
4.2.6: Pipe Blade Overview This tutorial example uses the Collapse function to create a degenerate topology in a Conjugate Heat transfer problem around a blade located in the center of a cylindrical pipe.
More informationacute angle An angle with a measure less than that of a right angle. Houghton Mifflin Co. 2 Grade 5 Unit 6
acute angle An angle with a measure less than that of a right angle. Houghton Mifflin Co. 2 Grade 5 Unit 6 angle An angle is formed by two rays with a common end point. Houghton Mifflin Co. 3 Grade 5 Unit
More informationGeometry. Geometry. Domain Cluster Standard. Congruence (G CO)
Domain Cluster Standard 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance
More informationPearson Mathematics Geometry Common Core 2015
A Correlation of Pearson Mathematics Geometry Common Core 2015 to the Common Core State Standards for Bid Category 13-040-10 A Correlation of Pearson, Common Core Pearson Geometry Congruence G-CO Experiment
More informationMathematics Standards for High School Geometry
Mathematics Standards for High School Geometry Geometry is a course required for graduation and course is aligned with the College and Career Ready Standards for Mathematics in High School. Throughout
More informationGeometry/Pre AP Geometry Common Core Standards
1st Nine Weeks Transformations Transformations *Rotations *Dilation (of figures and lines) *Translation *Flip G.CO.1 Experiment with transformations in the plane. Know precise definitions of angle, circle,
More informationGEOMETRY Curriculum Overview
GEOMETRY Curriculum Overview Semester 1 Semester 2 Unit 1 ( 5 1/2 Weeks) Unit 2 Unit 3 (2 Weeks) Unit 4 (1 1/2 Weeks) Unit 5 (Semester Break Divides Unit) Unit 6 ( 2 Weeks) Unit 7 (7 Weeks) Lines and Angles,
More information10.1 Overview. Section 10.1: Overview. Section 10.2: Procedure for Generating Prisms. Section 10.3: Prism Meshing Options
Chapter 10. Generating Prisms This chapter describes the automatic and manual procedure for creating prisms in TGrid. It also discusses the solution to some common problems that you may face while creating
More informationStandards to Topics. Common Core State Standards 2010 Geometry
Standards to Topics G-CO.01 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance
More informationTest #1: Chapters 1, 2, 3 Test #2: Chapters 4, 7, 9 Test #3: Chapters 5, 6, 8 Test #4: Chapters 10, 11, 12
Progress Assessments When the standards in each grouping are taught completely the students should take the assessment. Each assessment should be given within 3 days of completing the assigned chapters.
More informationCommon Core Specifications for Geometry
1 Common Core Specifications for Geometry Examples of how to read the red references: Congruence (G-Co) 2-03 indicates this spec is implemented in Unit 3, Lesson 2. IDT_C indicates that this spec is implemented
More informationGeometry. Cluster: Experiment with transformations in the plane. G.CO.1 G.CO.2. Common Core Institute
Geometry Cluster: Experiment with transformations in the plane. G.CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of
More informationAppendix E. Plane Geometry
Appendix E Plane Geometry A. Circle A circle is defined as a closed plane curve every point of which is equidistant from a fixed point within the curve. Figure E-1. Circle components. 1. Pi In mathematics,
More informationHonors Geometry Pacing Guide Honors Geometry Pacing First Nine Weeks
Unit Topic To recognize points, lines and planes. To be able to recognize and measure segments and angles. To classify angles and name the parts of a degree To recognize collinearity and betweenness of
More informationYEAR AT A GLANCE Student Learning Outcomes by Marking Period
2014-2015 Term 1 Overarching/general themes: Tools to Build and Analyze Points, Lines and Angles Dates Textual References To Demonstrate Proficiency by the End of the Term Students Will : Marking Period
More informationMADISON ACADEMY GEOMETRY PACING GUIDE
MADISON ACADEMY GEOMETRY PACING GUIDE 2018-2019 Standards (ACT included) ALCOS#1 Know the precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined
More informationRussell County Pacing Guide
August Experiment with transformations in the plane. 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance
More informationNote: For all questions, answer (E) NOTA means none of the above answers is correct. Unless otherwise specified, all angles are measured in degrees.
Note: For all questions, answer means none of the above answers is correct. Unless otherwise specified, all angles are measured in degrees. 1. The three angles of a triangle have measures given by 3 5,
More informationGeometry I Can Statements I can describe the undefined terms: point, line, and distance along a line in a plane I can describe the undefined terms:
Geometry I Can Statements I can describe the undefined terms: point, line, and distance along a line in a plane I can describe the undefined terms: point, line, and distance along a line in a plane I can
More informationGeometry Critical Areas of Focus
Ohio s Learning Standards for Mathematics include descriptions of the Conceptual Categories. These descriptions have been used to develop critical areas for each of the courses in both the Traditional
More information2. MODELING A MIXING ELBOW (2-D)
MODELING A MIXING ELBOW (2-D) 2. MODELING A MIXING ELBOW (2-D) In this tutorial, you will use GAMBIT to create the geometry for a mixing elbow and then generate a mesh. The mixing elbow configuration is
More informationNFC ACADEMY COURSE OVERVIEW
NFC ACADEMY COURSE OVERVIEW Geometry Honors is a full year, high school math course for the student who has successfully completed the prerequisite course, Algebra I. The course focuses on the skills and
More informationHS Geometry Mathematics CC
Course Description This course involves the integration of logical reasoning and spatial visualization skills. It includes a study of deductive proofs and applications from Algebra, an intense study of
More informationYou can read a TGrid mesh file using the File/Read/Mesh... menu item or the text command file/read-mesh.
Appendix E. Tips This appendix contains tips on the following topics: Section E.1: Reading Files Section E.2: Writing Files Section E.3: Saving Hard Copy Files Section E.4: Importing Meshes Section E.5:
More informationChapter 10 Similarity
Chapter 10 Similarity Def: The ratio of the number a to the number b is the number. A proportion is an equality between ratios. a, b, c, and d are called the first, second, third, and fourth terms. The
More informationSurface Mesh Generation
Surface Mesh Generation J.-F. Remacle Université catholique de Louvain September 22, 2011 0 3D Model For the description of the mesh generation process, let us consider the CAD model of a propeller presented
More informationAchievement Level Descriptors Geometry
Achievement Level Descriptors Geometry ALD Stard Level 2 Level 3 Level 4 Level 5 Policy MAFS Students at this level demonstrate a below satisfactory level of success with the challenging Students at this
More informationGeometry Common Core State Standard (CCSS) Math
= ntroduced R=Reinforced/Reviewed HGH SCHOOL GEOMETRY MATH STANDARDS 1 2 3 4 Congruence Experiment with transformations in the plane G.CO.1 Know precise definitions of angle, circle, perpendicular line,
More informationSOL Chapter Due Date
Name: Block: Date: Geometry SOL Review SOL Chapter Due Date G.1 2.2-2.4 G.2 3.1-3.5 G.3 1.3, 4.8, 6.7, 9 G.4 N/A G.5 5.5 G.6 4.1-4.7 G.7 6.1-6.6 G.8 7.1-7.7 G.9 8.2-8.6 G.10 1.6, 8.1 G.11 10.1-10.6, 11.5,
More informationMathematics Geometry
Common Core Correlations Mathematics Geometry Please note the following abbreviations found in this document: A=Activity L=Lesson AP=Activity Practice EA=Embedded Assessment GR=Getting Ready BENCHMARK
More informationUnit Activity Correlations to Common Core State Standards. Geometry. Table of Contents. Geometry 1 Statistics and Probability 8
Unit Activity Correlations to Common Core State Standards Geometry Table of Contents Geometry 1 Statistics and Probability 8 Geometry Experiment with transformations in the plane 1. Know precise definitions
More informationGeometry SEMESTER 1 SEMESTER 2
SEMESTER 1 Geometry 1. Geometry Basics 2. Coordinate Geometry a) Transformations, e.g., T(x + a, y + b) 3. Angles 4. Triangles a) Circumcenter 5. Construction a) Copy a segment, angle b) Bisect a segment,
More informationOhio s Learning Standards Mathematics Scope and Sequence YEC Geometry
Youngstown City School District English Language Arts Scope and Sequence Grade K Ohio s Learning Standards Mathematics Scope and Sequence YEC Geometry Mathematics Standards Scope and Sequence, YEC Geometry
More information7. 2 More Things Under. Construction. A Develop Understanding Task
7 Construction A Develop Understanding Task Like a rhombus, an equilateral triangle has three congruent sides. Show and describe how you might locate the third vertex point on an equilateral triangle,
More informationOptimizing triangular meshes to have the incrircle packing property
Packing Circles and Spheres on Surfaces Ali Mahdavi-Amiri Introduction Optimizing triangular meshes to have p g g the incrircle packing property Our Motivation PYXIS project Geometry Nature Geometry Isoperimetry
More informationSmarter Balanced Vocabulary (from the SBAC test/item specifications)
Example: Smarter Balanced Vocabulary (from the SBAC test/item specifications) Notes: Most terms area used in multiple grade levels. You should look at your grade level and all of the previous grade levels.
More informationGeometry Spring 2017 Item Release
Geometry Spring 2017 Item Release 1 Geometry Reporting Category: Congruence and Proof Question 2 16743 20512 Content Cluster: Use coordinates to prove simple geometric theorems algebraically and to verify
More informationThe radius for a regular polygon is the same as the radius of the circumscribed circle.
Perimeter and Area The perimeter and area of geometric shapes are basic properties that we need to know. The more complex a shape is, the more complex the process can be in finding its perimeter and area.
More informationSelect the best answer. Bubble the corresponding choice on your scantron. Team 13. Geometry
Team Geometry . What is the sum of the interior angles of an equilateral triangle? a. 60 b. 90 c. 80 d. 60. The sine of angle A is. What is the cosine of angle A? 6 4 6 a. b. c.. A parallelogram has all
More informationGeometry. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Common Core State for Mathematics High School Following is a correlation of Pearson s Prentice Hall Common Core Geometry 2012 to Common Core State for High School Mathematics. Geometry Congruence G-CO
More informationGrade 9, 10 or 11- Geometry
Grade 9, 10 or 11- Geometry Strands 1. Congruence, Proof, and Constructions 2. Similarity, Proof, and Trigonometry 3. Extending to Three Dimensions 4. Connecting Algebra and Geometry through Coordinates
More informationGEOMETRY Graded Course of Study
GEOMETRY Graded Course of Study Conceptual Category: Domain: Congruence Experiment with transformations in the plane. Understand congruence in terms of rigid motions. Prove geometric theorems both formally
More informationGrade VIII. Mathematics Geometry Notes. #GrowWithGreen
Grade VIII Mathematics Geometry Notes #GrowWithGreen Polygons can be classified according to their number of sides (or vertices). The sum of all the interior angles of an n -sided polygon is given by,
More informationGeometry Vocabulary Math Fundamentals Reference Sheet Page 1
Math Fundamentals Reference Sheet Page 1 Acute Angle An angle whose measure is between 0 and 90 Acute Triangle A that has all acute Adjacent Alternate Interior Angle Two coplanar with a common vertex and
More informationGeometry Assessment Structure for Mathematics:
Geometry Assessment Structure for 2013-2014 Mathematics: The Common Core State Standards for Mathematics are organized into Content Standards which define what students should understand and be able to
More informationGEOMETRY. Changes to the original 2010 COS is in red. If it is red and crossed out, it has been moved to another course.
The Geometry course builds on Algebra I concepts and increases students knowledge of shapes and their properties through geometry-based applications, many of which are observable in aspects of everyday
More informationGeometry GEOMETRY. Congruence
Geometry Geometry builds on Algebra I concepts and increases students knowledge of shapes and their properties through geometry-based applications, many of which are observable in aspects of everyday life.
More informationGEOMETRY is the study of points in space
CHAPTER 5 Logic and Geometry SECTION 5-1 Elements of Geometry GEOMETRY is the study of points in space POINT indicates a specific location and is represented by a dot and a letter R S T LINE is a set of
More informationGuide Assessment Structure Geometry
Guide Assessment Structure Geometry The Common Core State Standards for Mathematics are organized into Content Standards which define what students should understand and be able to do. Related standards
More informationNEW YORK GEOMETRY TABLE OF CONTENTS
NEW YORK GEOMETRY TABLE OF CONTENTS CHAPTER 1 POINTS, LINES, & PLANES {G.G.21, G.G.27} TOPIC A: Concepts Relating to Points, Lines, and Planes PART 1: Basic Concepts and Definitions...1 PART 2: Concepts
More informationHoughton Mifflin Harcourt Geometry 2015 correlated to the New York Common Core Learning Standards for Mathematics Geometry
Houghton Mifflin Harcourt Geometry 2015 correlated to the New York Common Core Learning Standards for Mathematics Geometry Standards for Mathematical Practice SMP.1 Make sense of problems and persevere
More informationParallel Computation of Spherical Parameterizations for Mesh Analysis. Th. Athanasiadis and I. Fudos University of Ioannina, Greece
Parallel Computation of Spherical Parameterizations for Mesh Analysis Th. Athanasiadis and I. Fudos, Greece Introduction Mesh parameterization is a powerful geometry processing tool Applications Remeshing
More informationCCSD Proficiency Scale - Language of Geometry
CCSD Scale - Language of Geometry Content Area: HS Math Grade : Geometry Standard Code: G-CO.1 application G-CO.1 Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line
More informationOhio s Learning Standards-Extended. Mathematics. Congruence Standards Complexity a Complexity b Complexity c
Ohio s Learning Standards-Extended Mathematics Congruence Standards Complexity a Complexity b Complexity c Most Complex Least Complex Experiment with transformations in the plane G.CO.1 Know precise definitions
More informationOhio Tutorials are designed specifically for the Ohio Learning Standards to prepare students for the Ohio State Tests and end-ofcourse
Tutorial Outline Ohio Tutorials are designed specifically for the Ohio Learning Standards to prepare students for the Ohio State Tests and end-ofcourse exams. Math Tutorials offer targeted instruction,
More information2003/2010 ACOS MATHEMATICS CONTENT CORRELATION GEOMETRY 2003 ACOS 2010 ACOS
CURRENT ALABAMA CONTENT PLACEMENT G.1 Determine the equation of a line parallel or perpendicular to a second line through a given point. G.2 Justify theorems related to pairs of angles, including angles
More informationIntroduction to Geometry
Introduction to Geometry This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (211 topics + 6 additional topics)
More informationGEOMETRY CURRICULUM MAP
2017-2018 MATHEMATICS GEOMETRY CURRICULUM MAP Department of Curriculum and Instruction RCCSD Congruence Understand congruence in terms of rigid motions Prove geometric theorems Common Core Major Emphasis
More informationAcknowledgement: Scott, Foresman. Geometry. SIMILAR TRIANGLES. 1. Definition: A ratio represents the comparison of two quantities.
1 cknowledgement: Scott, Foresman. Geometry. SIMILR TRINGLS 1. efinition: ratio represents the comparison of two quantities. In figure, ratio of blue squares to white squares is 3 : 5 2. efinition: proportion
More informationCarnegie Learning High School Math Series: Geometry Indiana Standards Worktext Correlations
Carnegie Learning High School Math Series: Logic and Proofs G.LP.1 Understand and describe the structure of and relationships within an axiomatic system (undefined terms, definitions, axioms and postulates,
More informationUnit Number of Days Dates. 1 Angles, Lines and Shapes 14 8/2 8/ Reasoning and Proof with Lines and Angles 14 8/22 9/9
8 th Grade Geometry Curriculum Map Overview 2016-2017 Unit Number of Days Dates 1 Angles, Lines and Shapes 14 8/2 8/19 2 - Reasoning and Proof with Lines and Angles 14 8/22 9/9 3 - Congruence Transformations
More information(1) Find the area of an equilateral triangle if each side is 8. (2) Given the figure to the right with measures as marked, find: mab, m BAF, m.
(1) ind the area of an equilateral triangle if each side is 8. (2) Given the figure to the right with measures as marked, find: m, m, m, m 100 9 90 (3) ind the length of the arc of a sector of in a circle
More informationDistrict 200 Geometry (I, A) Common Core Curriculum
Length: Two Semesters Prerequisite: Algebra 1 or equivalent District 200 Geometry (I, A) Common Core Curriculum How to read this document: CC.9-12.N.Q.2 Reason quantitatively and use units to solve problems.
More informationGiven a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G- CO.1 Identify Definitions Standard 1 Experiment with transformations in the plane. Know precise definitions of angle, circle, perpendicular line, parallel line, or line segment, based on the undefined
More informationPerformance Objectives Develop dictionary terms and symbols
Basic Geometry Course Name: Geometry Unit: 1 Terminology & Fundamental Definitions Time Line: 4 to 6 weeks Building Blocks of Geometry Define & identify point, line, plane angle, segment, ray, midpoint,
More informationBeal City High School Geometry Curriculum and Alignment
Beal City High School Geometry Curriculum and Alignment UNIT 1 Geometry Basics (Chapter 1) 1. Points, lines and planes (1-1, 1-2) 2. Axioms (postulates), theorems, definitions (Ch 1) 3. Angles (1-3) 4.
More informationKillingly Public Schools. Grades Draft Sept. 2002
Killingly Public Schools Grades 10-12 Draft Sept. 2002 ESSENTIALS OF GEOMETRY Grades 10-12 Language of Plane Geometry CONTENT STANDARD 10-12 EG 1: The student will use the properties of points, lines,
More informationMathematics High School Geometry An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts
Mathematics High School Geometry An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts interpreting a schematic drawing, estimating the amount of
More informationPearson Geometry Common Core 2015
A Correlation of Geometry Common Core to the Common Core State Standards for Mathematics High School , Introduction This document demonstrates how meets the Mathematics High School, PARRC Model Content
More information1 Automatic Mesh Generation
1 AUTOMATIC MESH GENERATION 1 1 Automatic Mesh Generation 1.1 Mesh Definition Mesh M is a discrete representation of geometric model in terms of its geometry G, topology T, and associated attributes A.
More informationUnit 1: Tools of Geometry
Unit 1: Tools of Geometry Geometry CP Pacing Guide First Nine Weeks Tennessee State Math Standards Know precise definitions of angle, circle, perpendicular line, parallel G.CO.A.1 line, and line segment,
More informationSequence of Geometry Modules Aligned with the Standards
Sequence of Geometry Modules Aligned with the Standards Module 1: Congruence, Proof, and Constructions Module 2: Similarity, Proof, and Trigonometry Module 3: Extending to Three Dimensions Module 4: Connecting
More informationViscous Hybrid Mesh Generation
Tutorial 4. Viscous Hybrid Mesh Generation Introduction In cases where you want to resolve the boundary layer, it is often more efficient to use prismatic cells in the boundary layer rather than tetrahedral
More informationGEOMETRY CCR MATH STANDARDS
CONGRUENCE, PROOF, AND CONSTRUCTIONS M.GHS. M.GHS. M.GHS. GEOMETRY CCR MATH STANDARDS Mathematical Habits of Mind. Make sense of problems and persevere in solving them.. Use appropriate tools strategically..
More informationGeometry. Instructional Activities:
GEOMETRY Instructional Activities: Geometry Assessment: A. Direct Instruction A. Quizzes B. Cooperative Learning B. Skill Reviews C. Technology Integration C. Test Prep Questions D. Study Guides D. Chapter
More informationMath Lesson Plan 6th Grade Curriculum Total Activities: 302
TimeLearning Online Learning for Homeschool and Enrichment www.timelearning.com Languages Arts, Math and more Multimedia s, Interactive Exercises, Printable Worksheets and Assessments Student Paced Learning
More informationCommon Core State Standards for Mathematics High School
Using the Program for Success Common Core State Standards for Mathematics High School The following shows the High School Standards for Mathematical Content that are taught in Pearson Common Core Edition
More information