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class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"github_copilot","context":"product","tag":"link","label":"github_copilot_link_product_navbar"}" href="https://github.com/features/copilot"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-copilot color-fg-subtle mr-3"> <path d="M23.922 16.992c-.861 1.495-5.859 5.023-11.922 5.023-6.063 0-11.061-3.528-11.922-5.023A.641.641 0 0 1 0 16.736v-2.869a.841.841 0 0 1 .053-.22c.372-.935 1.347-2.292 2.605-2.656.167-.429.414-1.055.644-1.517a10.195 10.195 0 0 1-.052-1.086c0-1.331.282-2.499 1.132-3.368.397-.406.89-.717 1.474-.952 1.399-1.136 3.392-2.093 6.122-2.093 2.731 0 4.767.957 6.166 2.093.584.235 1.077.546 1.474.952.85.869 1.132 2.037 1.132 3.368 0 .368-.014.733-.052 1.086.23.462.477 1.088.644 1.517 1.258.364 2.233 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14.25a1 1 0 0 1 1 1v2a1 1 0 0 1-2 0v-2a1 1 0 0 1 1-1Zm-5 0a1 1 0 0 1 1 1v2a1 1 0 0 1-2 0v-2a1 1 0 0 1 1-1Z"></path> </svg> <div> <div class="color-fg-default h4">GitHub Copilot</div> Write better code with AI </div> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"security","context":"product","tag":"link","label":"security_link_product_navbar"}" href="https://github.com/features/security"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-shield-check color-fg-subtle mr-3"> <path d="M16.53 9.78a.75.75 0 0 0-1.06-1.06L11 13.19l-1.97-1.97a.75.75 0 0 0-1.06 1.06l2.5 2.5a.75.75 0 0 0 1.06 0l5-5Z"></path><path d="m12.54.637 8.25 2.675A1.75 1.75 0 0 1 22 4.976V10c0 6.19-3.771 10.704-9.401 12.83a1.704 1.704 0 0 1-1.198 0C5.77 20.705 2 16.19 2 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1 2-2H21a2 2 0 0 1 2 2V21a2 2 0 0 1-2 2h-6.5a2 2 0 0 1-2-2v-2.5H8.437A2.939 2.939 0 0 1 5.5 15.562V11.5H3a2 2 0 0 1-2-2Zm2-.5a.5.5 0 0 0-.5.5v6.5a.5.5 0 0 0 .5.5h6.5a.5.5 0 0 0 .5-.5V3a.5.5 0 0 0-.5-.5ZM14.5 14a.5.5 0 0 0-.5.5V21a.5.5 0 0 0 .5.5H21a.5.5 0 0 0 .5-.5v-6.5a.5.5 0 0 0-.5-.5Z"></path> </svg> <div> <div class="color-fg-default h4">Actions</div> Automate any workflow </div> </a></li> <li> <a class="HeaderMenu-dropdown-link d-block no-underline position-relative py-2 Link--secondary d-flex flex-items-center Link--has-description pb-lg-3" data-analytics-event="{"location":"navbar","action":"codespaces","context":"product","tag":"link","label":"codespaces_link_product_navbar"}" href="https://github.com/features/codespaces"> <svg aria-hidden="true" height="24" viewBox="0 0 24 24" version="1.1" width="24" data-view-component="true" class="octicon octicon-codespaces color-fg-subtle mr-3"> <path d="M3.5 3.75C3.5 2.784 4.284 2 5.25 2h13.5c.966 0 1.75.784 1.75 1.75v7.5A1.75 1.75 0 0 1 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data-action="close:qbsearch-input#handleDialogClose cancel:qbsearch-input#handleDialogClose" id="feedback-dialog" aria-modal="true" aria-labelledby="feedback-dialog-title" aria-describedby="feedback-dialog-description" data-view-component="true" class="Overlay Overlay-whenNarrow Overlay--size-medium Overlay--motion-scaleFade Overlay--disableScroll"> <div data-view-component="true" class="Overlay-header"> <div class="Overlay-headerContentWrap"> <div class="Overlay-titleWrap"> <h1 class="Overlay-title " id="feedback-dialog-title"> Provide feedback </h1> </div> <div class="Overlay-actionWrap"> <button data-close-dialog-id="feedback-dialog" aria-label="Close" type="button" data-view-component="true" class="close-button Overlay-closeButton"><svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-x"> <path d="M3.72 3.72a.75.75 0 0 1 1.06 0L8 6.94l3.22-3.22a.749.749 0 0 1 1.275.326.749.749 0 0 1-.215.734L9.06 8l3.22 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Overlay--size-medium Overlay--motion-scaleFade Overlay--disableScroll"> <div data-view-component="true" class="Overlay-header Overlay-header--divided"> <div class="Overlay-headerContentWrap"> <div class="Overlay-titleWrap"> <h1 class="Overlay-title " id="custom-scopes-dialog-title"> Saved searches </h1> <h2 id="custom-scopes-dialog-description" class="Overlay-description">Use saved searches to filter your results more quickly</h2> </div> <div class="Overlay-actionWrap"> <button data-close-dialog-id="custom-scopes-dialog" aria-label="Close" type="button" data-view-component="true" class="close-button Overlay-closeButton"><svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-x"> <path d="M3.72 3.72a.75.75 0 0 1 1.06 0L8 6.94l3.22-3.22a.749.749 0 0 1 1.275.326.749.749 0 0 1-.215.734L9.06 8l3.22 3.22a.749.749 0 0 1-.326 1.275.749.749 0 0 1-.734-.215L8 9.06l-3.22 3.22a.751.751 0 0 1-1.042-.018.751.751 0 0 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src="/search/custom_scopes/check_name" required> <input type="text" name="custom_scope_name" id="custom_scope_name" data-target="custom-scopes.customScopesNameField" class="form-control" autocomplete="off" placeholder="github-ruby" required maxlength="50"> <input type="hidden" data-csrf="true" value="27KZQ0sV1IH0G+xJr9wowN4znvv6TiTJybCgAZjIK8NLzIOduHiiRMCEz2JNbDBHouwChc7RC8FmCdzXk5NV8g==" /> </auto-check> </div> <div class="form-group"> <label for="custom_scope_query">Query</label> <input type="text" name="custom_scope_query" id="custom_scope_query" data-target="custom-scopes.customScopesQueryField" class="form-control" autocomplete="off" placeholder="(repo:mona/a OR repo:mona/b) AND lang:python" required maxlength="500"> </div> <p class="text-small color-fg-muted"> To see all available qualifiers, see our <a class="Link--inTextBlock" href="https://docs.github.com/search-github/github-code-search/understanding-github-code-search-syntax">documentation</a>. </p> </form> </div> <div 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rounded-lg-0 px-2 py-1" style="margin-left: 12px;" data-hydro-click="{"event_type":"authentication.click","payload":{"location_in_page":"site header menu","repository_id":null,"auth_type":"SIGN_UP","originating_url":"https://github.com/austinEng/WebGL-PIC-FLIP-Fluid","user_id":null}}" data-hydro-click-hmac="286722f8680682d31a1c983504c6b5a13942898751fbe28784842f8637aba308" data-analytics-event="{"category":"Marketing nav","action":"click to go to homepage","label":"ref_page:Marketing;ref_cta:Sign in;ref_loc:Header"}" > Sign in </a> </div> <a href="/signup?ref_cta=Sign+up&ref_loc=header+logged+out&ref_page=%2F%3Cuser-name%3E%2F%3Crepo-name%3E&source=header-repo&source_repo=austinEng%2FWebGL-PIC-FLIP-Fluid" class="HeaderMenu-link HeaderMenu-link--sign-up HeaderMenu-button flex-shrink-0 d-flex d-lg-inline-flex no-underline border color-border-default rounded px-2 py-1" data-hydro-click="{"event_type":"authentication.click","payload":{"location_in_page":"site header menu","repository_id":null,"auth_type":"SIGN_UP","originating_url":"https://github.com/austinEng/WebGL-PIC-FLIP-Fluid","user_id":null}}" data-hydro-click-hmac="286722f8680682d31a1c983504c6b5a13942898751fbe28784842f8637aba308" data-analytics-event="{"category":"Sign up","action":"click to sign up for account","label":"ref_page:/<user-name>/<repo-name>;ref_cta:Sign up;ref_loc:header logged out"}" > Sign up </a> <button type="button" class="sr-only js-header-menu-focus-trap d-block d-lg-none">Reseting focus</button> </div> </div> </div> </div> </header> <div hidden="hidden" data-view-component="true" class="js-stale-session-flash stale-session-flash flash flash-warn flash-full"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-alert"> <path d="M6.457 1.047c.659-1.234 2.427-1.234 3.086 0l6.082 11.378A1.75 1.75 0 0 1 14.082 15H1.918a1.75 1.75 0 0 1-1.543-2.575Zm1.763.707a.25.25 0 0 0-.44 0L1.698 13.132a.25.25 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1-2.25 2.25h-1.5v2.128a2.251 2.251 0 1 1-1.5 0V8.5h-1.5A2.25 2.25 0 0 1 3.5 6.25v-.878a2.25 2.25 0 1 1 1.5 0ZM5 3.25a.75.75 0 1 0-1.5 0 .75.75 0 0 0 1.5 0Zm6.75.75a.75.75 0 1 0 0-1.5.75.75 0 0 0 0 1.5Zm-3 8.75a.75.75 0 1 0-1.5 0 .75.75 0 0 0 1.5 0Z"></path> </svg>Fork <span id="repo-network-counter" data-pjax-replace="true" data-turbo-replace="true" title="11" data-view-component="true" class="Counter">11</span> </a> </li> <li> <div data-view-component="true" class="BtnGroup d-flex"> <a href="/login?return_to=%2FaustinEng%2FWebGL-PIC-FLIP-Fluid" rel="nofollow" data-hydro-click="{"event_type":"authentication.click","payload":{"location_in_page":"star button","repository_id":74245659,"auth_type":"LOG_IN","originating_url":"https://github.com/austinEng/WebGL-PIC-FLIP-Fluid","user_id":null}}" data-hydro-click-hmac="37f125202877c8799ac12a287f40685e3982abfaf0f06c7f906a7110e498963f" aria-label="You must be signed in to star a repository" data-view-component="true" class="tooltipped 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data-view-component="true" class="Counter js-social-count">179</span> </a></div> </li> </ul> </div> </div> <div id="responsive-meta-container" data-turbo-replace> <div class="d-block d-md-none mb-2 px-3 px-md-4 px-lg-5"> <p class="f4 mb-3 "> PIC/FLIP Fluid with WebGL </p> <div class="mb-2 d-flex flex-items-center Link--secondary"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-link flex-shrink-0 mr-2"> <path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path> </svg> <span class="flex-auto min-width-0 css-truncate css-truncate-target width-fit"> <a title="http://austineng.github.io/WebGL-PIC-FLIP-Fluid/demo/" role="link" target="_blank" class="text-bold" rel="noopener noreferrer" href="http://austineng.github.io/WebGL-PIC-FLIP-Fluid/demo/">austineng.github.io/webgl-pic-flip-fluid/demo/</a> </span> </div> <div class="mb-3"> <a class="Link--secondary no-underline mr-3" href="/austinEng/WebGL-PIC-FLIP-Fluid/stargazers"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" version="1.1" width="16" data-view-component="true" class="octicon octicon-star mr-1"> <path d="M8 .25a.75.75 0 0 1 .673.418l1.882 3.815 4.21.612a.75.75 0 0 1 .416 1.279l-3.046 2.97.719 4.192a.751.751 0 0 1-1.088.791L8 12.347l-3.766 1.98a.75.75 0 0 1-1.088-.79l.72-4.194L.818 6.374a.75.75 0 0 1 .416-1.28l4.21-.611L7.327.668A.75.75 0 0 1 8 .25Zm0 2.445L6.615 5.5a.75.75 0 0 1-.564.41l-3.097.45 2.24 2.184a.75.75 0 0 1 .216.664l-.528 3.084 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class=\"heading-element\" dir=\"auto\"\u003eWebGL PIC/FLIP Fluid\u003c/h1\u003e\u003ca id=\"user-content-webgl-picflip-fluid\" class=\"anchor\" aria-label=\"Permalink: WebGL PIC/FLIP Fluid\" href=\"#webgl-picflip-fluid\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eby \u003ca href=\"http://austineng.github.io\" rel=\"nofollow\"\u003eAustin Eng\u003c/a\u003e\u003c/p\u003e\n\u003col dir=\"auto\"\u003e\n\u003cli\u003e\u003ca href=\"#overview\"\u003eOverview\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"#debug-views\"\u003eDebug Views\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"#techniques\"\u003eTechniques\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"#performance\"\u003ePerformance\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003e\u003ca href=\"#build\"\u003eBuild \u0026amp; Running\u003c/a\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eOverview\u003c/h2\u003e\u003ca id=\"user-content-overview\" class=\"anchor\" aria-label=\"Permalink: Overview\" href=\"#overview\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThis project is a WebGL fluid simulation implementing of the PIC/FLIP method described in Zhu \u0026amp; Bridson's Siggraph 2005 paper \u003ca href=\"https://www.cs.ubc.ca/~rbridson/docs/zhu-siggraph05-sandfluid.pdf\" rel=\"nofollow\"\u003eAnimating Sand as a Fluid\u003c/a\u003e. Bridson's book \u003cem\u003eFluid Simulation for Computer Graphics\u003c/em\u003e was also an invaluable resource while working on this project. For a better understanding of these techniques, it is definitely advised to at least read those two resources.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003e\u003ca href=\"http://austineng.github.io/WebGL-PIC-FLIP-Fluid/demo/\" rel=\"nofollow\"\u003eLive Demo\u003c/a\u003e\u003c/h2\u003e\u003ca id=\"user-content-live-demo\" class=\"anchor\" aria-label=\"Permalink: Live Demo\" href=\"#live-demo\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer\" href=\"/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/fluid.gif\"\u003e\u003cimg src=\"/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/fluid.gif\" alt=\"\" data-animated-image=\"\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eDebug Views\u003c/h2\u003e\u003ca id=\"user-content-debug-views\" class=\"anchor\" aria-label=\"Permalink: Debug Views\" href=\"#debug-views\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer\" href=\"/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/velocity-debug.png\"\u003e\u003cimg src=\"/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/velocity-debug.png\" alt=\"Velocity\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\nDebug view showing grid velocities\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer\" href=\"/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/type-debug.png\"\u003e\u003cimg src=\"/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/type-debug.png\" alt=\"Type\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\nDebug view showing grid cell types\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer\" href=\"/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/divergence-debug.png\"\u003e\u003cimg src=\"/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/divergence-debug.png\" alt=\"Divergence\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\nDebug view showing grid divergences\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer\" href=\"/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/pressure-debug.png\"\u003e\u003cimg src=\"/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/pressure-debug.png\" alt=\"Pressure\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\nDebug view showing grid pressures\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003ca target=\"_blank\" rel=\"noopener noreferrer\" href=\"/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/value-debug.png\"\u003e\u003cimg src=\"/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/value-debug.png\" alt=\"Value\" style=\"max-width: 100%;\"\u003e\u003c/a\u003e\nDebug view showing values of the pressure solve. All steps of the conjugate gradient solve are debuggable which was invaluable in checking the math.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eTechniques\u003c/h2\u003e\u003ca id=\"user-content-techniques\" class=\"anchor\" aria-label=\"Permalink: Techniques\" href=\"#techniques\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eAs the name implies, PIC/FLIP is a hybrid method which mixes the perspectives of solving the system from a particle point of view (Lagrangian) and solving the system from a grid point of view (Eulerian). One of the motivations behind this is that it is much easier to solve for forces such as pressure on a uniform grid, but it makes more sense to keep track of specific particle attributes such as position and velocity on the particles themselves. Thus, for every frame, we project our particle attributes onto the grid, compute forces and presssure on the grid, and then interpolate updated velocities back onto the particles before advection. The PIC/FLIP part comes in how we update with new velocities. With the PIC approach, we simply update particles with the new grid velocity. With the FLIP approach, we increment the particle velocity by the delta in grid velocity. The latter better maintains individual particle velocities but can become chaotic, so it is common to use a blend of 5% PIC and 95% FLIP.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe following will be an abbreviated breakdown of how all of the steps in the fluid solve are accomplished in shaders.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eParticle-to-Grid\u003c/h3\u003e\u003ca id=\"user-content-particle-to-grid\" class=\"anchor\" aria-label=\"Permalink: Particle-to-Grid\" href=\"#particle-to-grid\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThis first step is a fairly tricky and annoying part of the solve. For every particle, we need to project it's velocity to possibly 3x27 = 81 different locations. There are three grids to project onto (U, V, W for X, Y, Z velocities) but we need to project onto the 27 cells in our direct neighborhood. To do this, we do exactly that. For every particle, we render 81 gl.POINTS. In practice, this is calling \u003ccode\u003egl.drawArrays(pointBuffer, 0, 81*pointCount)\u003c/code\u003e.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eWe number these points with increasing values so that in the vertex shader we can correctly compute which grid we are to project onto. After calculating the correct grid and grid location, we compute the UV coordinate of that position and set glPosition accordingly.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eNote that we want each grid cell to store the weighted average of all the nearby particles so we actually need to accumulate all of the nearby velocities and keep track of their weights. To do this, we preface our draw call with \u003ccode\u003egl.enable(gl.BLEND)\u003c/code\u003e and \u003ccode\u003egl.blendFunc(gl.ONE, gl.ONE)\u003c/code\u003e. This will allow us to take advantage of hardware blending functions to accumulate influences. In practice, we do this draw call twice, writing to different framebuffers. To one we write velocity * weight, and to the other we write just the weight. Finally, we bind both textures and render a texture-sized quad, dividing all accumulated velocities by their respective cumulative weight.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eCopy-Grid\u003c/h3\u003e\u003ca id=\"user-content-copy-grid\" class=\"anchor\" aria-label=\"Permalink: Copy-Grid\" href=\"#copy-grid\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eAt this point, we copy our velocity field to a new texture. This is important for later computing the change in grid velocity.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eMark-Cells\u003c/h3\u003e\u003ca id=\"user-content-mark-cells\" class=\"anchor\" aria-label=\"Permalink: Mark-Cells\" href=\"#mark-cells\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eIn this step we mark cells with the type of their contents: either SOLID, FLUID, or AIR. Marking cells is important because we use this information later to set up our pressure equations. To accomplish this, we do a similar scatter-approach taken in the particle-to-grid transfer. All cells are originally air (value is 0). We then render 7 points for each particle. We use 7 because we actually want to mark the cells at the fluid boundary as fluid as well, we we extend the fluid-marking in each cardinal direction. Using the point index, we compute the grid cell we want to project onto and write a value of 1.0 (FLUID) to the computed UV coordinate. If there are solids in the scene, we do the same thing for them. Note: we \u003cstrong\u003edo not\u003c/strong\u003e use \u003ccode\u003egl.blend(gl.ONE, gl.ONE)\u003c/code\u003e. We do not want to accumulate the values of cell types.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eGravity-Update\u003c/h3\u003e\u003ca id=\"user-content-gravity-update\" class=\"anchor\" aria-label=\"Permalink: Gravity-Update\" href=\"#gravity-update\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThis is a fairly simple step where we just render a fullscreen quad and increment the y-velocities by G * timestep. In practice, I only update velocities near non-air cells or the debug views become difficult to view.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003ePressure-solve\u003c/h3\u003e\u003ca id=\"user-content-pressure-solve\" class=\"anchor\" aria-label=\"Permalink: Pressure-solve\" href=\"#pressure-solve\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThis is the most computational intensive part of the fluid solve. This is fairly complicated (it is definitely advised to read Bridson's book), but in short, we want to solve a pressure equation that will result in 0 divergence within the fluid. In other words, we want to compute pressures that will cause the fluid to move in such a way that it does not lose or gain any volume.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThis comes down to solving a system \u003ccode\u003eAp = b\u003c/code\u003e, where \u003ccode\u003eb\u003c/code\u003e is a vector of divergences (total change in velocity across a cell), and \u003ccode\u003eA\u003c/code\u003e is a coefficient matrix. If we have an \u003ccode\u003eX x Y x Z\u003c/code\u003e grid, the dimensions of \u003ccode\u003eA\u003c/code\u003e are \u003ccode\u003eX*Y*Z x X*Y*Z\u003c/code\u003e. If you work through the pressure equations, we realize that the diagonal entries of A denote the number of non-solid neighbors that cell \u003ccode\u003e(i, j, k)\u003c/code\u003e has, and the entries of the form \u003ccode\u003e\u0026lt;(i1, j1, k1), (i2, j2, k2)\u0026gt;\u003c/code\u003e are \u003ccode\u003e-1\u003c/code\u003e if those two cells are both fluid and \u003ccode\u003e0\u003c/code\u003e otherwise. Fortunately, this means that \u003ccode\u003eA\u003c/code\u003e is a very, very sparse matrix because each cell has at most 6 neighbors. Every row and column in \u003ccode\u003eA\u003c/code\u003e has \u003cstrong\u003eat most\u003c/strong\u003e 7 entries. Furthermore, A is symmetric so half of the non-diagonal entries don't need to be stored. This ultimately comes down to just 4 entries per cell, which is easily stored in the RGBA components of a single texture!\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eThe system is solved using the \u003ca href=\"https://en.wikipedia.org/wiki/Conjugate_gradient_method#The_preconditioned_conjugate_gradient_method\" rel=\"nofollow\"\u003epreconditioned conjugate gradient method\u003c/a\u003e which is an iterative method for solving systems like this (symmetric, positive-definite). The details are better explained elsewhere but I will briefly explain how some of the important steps are accomplished.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003ePreconditioning\u003c/strong\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003ePreconditioning is an extremely important step of the solve which makes it signficiantly faster to solve. \u003ccode\u003eA\u003c/code\u003e is typically not an invertible matrix, but preconditioning constructs an approximate inverse of \u003ccode\u003eA\u003c/code\u003e which enables us to get much closer to the correct solution in less time. Typically, CPU solvers use an Incomplete Cholesky preconditioner which works very, very well. However, it is not well suited to a GPU or shader implementation because it requires doing two triangular matrix solves with forward and backsubstitution for every iteration. On a GPU this is absolutely terrible and would involve hundreds of serial kernel invocations or draw calls. Instead, we use the Incomplete Poisson preconditioner which is accomplished with just two draw calls.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eGiven a residual vector \u003ccode\u003er\u003c/code\u003e, we want compute \u003ccode\u003ez = Mr\u003c/code\u003e, where \u003ccode\u003eM\u003c/code\u003e is an approximate inverse of \u003ccode\u003eA\u003c/code\u003e. The Incomplete Poisson preconditioner says that M = (I - LD\u003csup\u003e-1\u003c/sup\u003e)(I - D\u003csup\u003e-1\u003c/sup\u003eL) where \u003ccode\u003eL\u003c/code\u003e i strictly the lower diagonal entries of \u003ccode\u003eA\u003c/code\u003e and \u003ccode\u003eD\u003c/code\u003e is strictly the diagonal entires of \u003ccode\u003eA\u003c/code\u003e. We just need to compute z = (I - LD\u003csup\u003e-1\u003c/sup\u003e)(I - D\u003csup\u003e-1\u003c/sup\u003eL)r which is just two draw calls, one for each matrix multiply.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eFor each entry in \u003ccode\u003er\u003c/code\u003e:\u003c/p\u003e\n\u003col dir=\"auto\"\u003e\n\u003cli\u003eSet \u003ccode\u003ez[i] = r[i] - r * L[i, :] * 1/d\u003c/code\u003e. This is easy because \u003ccode\u003eL\u003c/code\u003e is sparse and has just 3 items in row i. \u003ccode\u003ed\u003c/code\u003e is just the diagonal entry corresponding to \u003cem\u003erow\u003c/em\u003e i.\u003c/li\u003e\n\u003cli\u003eSet \u003ccode\u003ez[i] = z[i] - z * L[i, :] * 1/d\u003c/code\u003e. This is easy because \u003ccode\u003eL\u003c/code\u003e is sparse and has just 3 items in row i. \u003ccode\u003ed\u003c/code\u003e is just the diagonal entry corresponding to \u003cem\u003ecolumn\u003c/em\u003e i.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eDot Products\u003c/strong\u003e\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eUnfortunately, we need to take three dot products every iteration, which is not a very parallel operation. We once again take advantage of texture blending and using \u003ccode\u003egl.blend(gl.ONE, gl.ONE)\u003c/code\u003e and write all products to the same UV coordinate to accumulate them.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eVelocity Extrapolation\u003c/h3\u003e\u003ca id=\"user-content-velocity-extrapolation\" class=\"anchor\" aria-label=\"Permalink: Velocity Extrapolation\" href=\"#velocity-extrapolation\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eIn this step we extrapolate velocities to push the velocities one grid cell past the fluid cells. This is important so that particles don't lose energy when they come in contact with air cells. This is implemented in a fragment shader as a gather operation which does a few grid type checks in the cardinal directions and appropriately updates velocity.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch3 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eGrid-To-Particle\u003c/h3\u003e\u003ca id=\"user-content-grid-to-particle\" class=\"anchor\" aria-label=\"Permalink: Grid-To-Particle\" href=\"#grid-to-particle\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003eThe grid-to-particle transfer is much simpler than the particle-to-grid transfer. For this, we only need to compute our fractional grid cell index and then trilinear-ly interpolate our new updated velocity.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003ePerformance\u003c/h2\u003e\u003ca id=\"user-content-performance\" class=\"anchor\" aria-label=\"Permalink: Performance\" href=\"#performance\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003cp dir=\"auto\"\u003e\u003cstrong\u003eTested on: Windows 10, i7-4770K @ 3.50GHz 16GB, GTX 780 3072MB\u003c/strong\u003e\u003c/p\u003e\n\u003cmarkdown-accessiblity-table\u003e\u003ctable\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth\u003eGrid Density\u003c/th\u003e\n\u003cth\u003eFPS (Incomplete Cholesky)\u003c/th\u003e\n\u003cth\u003eFPS (Incomplete Poisson)\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd\u003e5,000\u003c/td\u003e\n\u003ctd\u003e40\u003c/td\u003e\n\u003ctd\u003e60\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003e10,000\u003c/td\u003e\n\u003ctd\u003e30\u003c/td\u003e\n\u003ctd\u003e60\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003e50,000\u003c/td\u003e\n\u003ctd\u003e9\u003c/td\u003e\n\u003ctd\u003e33\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003e100,000\u003c/td\u003e\n\u003ctd\u003e3\u003c/td\u003e\n\u003ctd\u003e17\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003e500,000\u003c/td\u003e\n\u003ctd\u003e0.5\u003c/td\u003e\n\u003ctd\u003e6\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003e1,000,000\u003c/td\u003e\n\u003ctd\u003e0.2\u003c/td\u003e\n\u003ctd\u003e3\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\u003c/markdown-accessiblity-table\u003e\n\u003cp dir=\"auto\"\u003eAs we can see, using the Incomplete Poisson preconditioner has signficant performance benefits and is able to more effectively handle high grid resolutions.\u003c/p\u003e\n\u003cp dir=\"auto\"\u003eKeeping grid density constant but doubling particle density, these numbers are exactly the same, confirming that the limiting factor for performance is the pressure solve, which is bound by grid density.\u003c/p\u003e\n\u003cdiv class=\"markdown-heading\" dir=\"auto\"\u003e\u003ch2 tabindex=\"-1\" class=\"heading-element\" dir=\"auto\"\u003eBuild\u003c/h2\u003e\u003ca id=\"user-content-build\" class=\"anchor\" aria-label=\"Permalink: Build\" href=\"#build\"\u003e\u003csvg class=\"octicon octicon-link\" viewBox=\"0 0 16 16\" version=\"1.1\" width=\"16\" height=\"16\" aria-hidden=\"true\"\u003e\u003cpath d=\"m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z\"\u003e\u003c/path\u003e\u003c/svg\u003e\u003c/a\u003e\u003c/div\u003e\n\u003col dir=\"auto\"\u003e\n\u003cli\u003e\u003ca href=\"https://nodejs.org/en/download/\" rel=\"nofollow\"\u003eInstall \u003ccode\u003eNode.js\u003c/code\u003e\u003c/a\u003e\u003c/li\u003e\n\u003cli\u003eFork and clone this repository.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003enpm install\u003c/code\u003e in the project root directory.\u003c/li\u003e\n\u003cli\u003e\u003ccode\u003enpm run build\u003c/code\u003e and open \u003ccode\u003edemo/index.html\u003c/code\u003e OR \u003ccode\u003enpm start\u003c/code\u003e and navigate to \u003ccode\u003elocalhost:3000/demo\u003c/code\u003e\u003c/li\u003e\n\u003c/ol\u003e\n\u003c/article\u003e","loaded":true,"timedOut":false,"errorMessage":null,"headerInfo":{"toc":[{"level":1,"text":"WebGL PIC/FLIP Fluid","anchor":"webgl-picflip-fluid","htmlText":"WebGL PIC/FLIP Fluid"},{"level":2,"text":"Overview","anchor":"overview","htmlText":"Overview"},{"level":2,"text":"Live Demo","anchor":"live-demo","htmlText":"Live Demo"},{"level":2,"text":"Debug Views","anchor":"debug-views","htmlText":"Debug Views"},{"level":2,"text":"Techniques","anchor":"techniques","htmlText":"Techniques"},{"level":3,"text":"Particle-to-Grid","anchor":"particle-to-grid","htmlText":"Particle-to-Grid"},{"level":3,"text":"Copy-Grid","anchor":"copy-grid","htmlText":"Copy-Grid"},{"level":3,"text":"Mark-Cells","anchor":"mark-cells","htmlText":"Mark-Cells"},{"level":3,"text":"Gravity-Update","anchor":"gravity-update","htmlText":"Gravity-Update"},{"level":3,"text":"Pressure-solve","anchor":"pressure-solve","htmlText":"Pressure-solve"},{"level":3,"text":"Velocity Extrapolation","anchor":"velocity-extrapolation","htmlText":"Velocity Extrapolation"},{"level":3,"text":"Grid-To-Particle","anchor":"grid-to-particle","htmlText":"Grid-To-Particle"},{"level":2,"text":"Performance","anchor":"performance","htmlText":"Performance"},{"level":2,"text":"Build","anchor":"build","htmlText":"Build"}],"siteNavLoginPath":"/login?return_to=https%3A%2F%2Fgithub.com%2FaustinEng%2FWebGL-PIC-FLIP-Fluid"}}],"overviewFilesProcessingTime":0}},"appPayload":{"helpUrl":"https://docs.github.com","findFileWorkerPath":"/assets-cdn/worker/find-file-worker-1583894afd38.js","findInFileWorkerPath":"/assets-cdn/worker/find-in-file-worker-67668e8c2caa.js","githubDevUrl":null,"enabled_features":{"code_nav_ui_events":false,"overview_shared_code_dropdown_button":false,"react_blob_overlay":false,"copilot_conversational_ux_embedding_update":false,"copilot_smell_icebreaker_ux":true,"copilot_workspace":false,"accessible_code_button":true}}}}</script> <div data-target="react-partial.reactRoot"><style data-styled="true" 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1.5h-8.5a.75.75 0 0 1 0-1.5ZM2 14a1 1 0 1 1 0-2 1 1 0 0 1 0 2Zm1-6a1 1 0 1 1-2 0 1 1 0 0 1 2 0ZM2 4a1 1 0 1 1 0-2 1 1 0 0 1 0 2Z"></path></svg></button></div><div class="Box-sc-g0xbh4-0 QkQOb js-snippet-clipboard-copy-unpositioned" data-hpc="true"><article class="markdown-body entry-content container-lg" itemprop="text"><div class="markdown-heading" dir="auto"><h1 tabindex="-1" class="heading-element" dir="auto">WebGL PIC/FLIP Fluid</h1><a id="user-content-webgl-picflip-fluid" class="anchor" aria-label="Permalink: WebGL PIC/FLIP Fluid" href="#webgl-picflip-fluid"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">by <a href="http://austineng.github.io" rel="nofollow">Austin Eng</a></p> <ol dir="auto"> <li><a href="#overview">Overview</a></li> <li><a href="#debug-views">Debug Views</a></li> <li><a href="#techniques">Techniques</a></li> <li><a href="#performance">Performance</a></li> <li><a href="#build">Build & Running</a></li> </ol> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Overview</h2><a id="user-content-overview" class="anchor" aria-label="Permalink: Overview" href="#overview"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">This project is a WebGL fluid simulation implementing of the PIC/FLIP method described in Zhu & Bridson's Siggraph 2005 paper <a href="https://www.cs.ubc.ca/~rbridson/docs/zhu-siggraph05-sandfluid.pdf" rel="nofollow">Animating Sand as a Fluid</a>. Bridson's book <em>Fluid Simulation for Computer Graphics</em> was also an invaluable resource while working on this project. For a better understanding of these techniques, it is definitely advised to at least read those two resources.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto"><a href="http://austineng.github.io/WebGL-PIC-FLIP-Fluid/demo/" rel="nofollow">Live Demo</a></h2><a id="user-content-live-demo" class="anchor" aria-label="Permalink: Live Demo" href="#live-demo"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto"><a target="_blank" rel="noopener noreferrer" href="/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/fluid.gif"><img src="/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/fluid.gif" alt="" data-animated-image="" style="max-width: 100%;"></a></p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Debug Views</h2><a id="user-content-debug-views" class="anchor" aria-label="Permalink: Debug Views" href="#debug-views"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto"><a target="_blank" rel="noopener noreferrer" href="/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/velocity-debug.png"><img src="/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/velocity-debug.png" alt="Velocity" style="max-width: 100%;"></a> Debug view showing grid velocities</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer" href="/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/type-debug.png"><img src="/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/type-debug.png" alt="Type" style="max-width: 100%;"></a> Debug view showing grid cell types</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer" href="/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/divergence-debug.png"><img src="/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/divergence-debug.png" alt="Divergence" style="max-width: 100%;"></a> Debug view showing grid divergences</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer" href="/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/pressure-debug.png"><img src="/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/pressure-debug.png" alt="Pressure" style="max-width: 100%;"></a> Debug view showing grid pressures</p> <p dir="auto"><a target="_blank" rel="noopener noreferrer" href="/austinEng/WebGL-PIC-FLIP-Fluid/blob/master/img/value-debug.png"><img src="/austinEng/WebGL-PIC-FLIP-Fluid/raw/master/img/value-debug.png" alt="Value" style="max-width: 100%;"></a> Debug view showing values of the pressure solve. All steps of the conjugate gradient solve are debuggable which was invaluable in checking the math.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Techniques</h2><a id="user-content-techniques" class="anchor" aria-label="Permalink: Techniques" href="#techniques"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">As the name implies, PIC/FLIP is a hybrid method which mixes the perspectives of solving the system from a particle point of view (Lagrangian) and solving the system from a grid point of view (Eulerian). One of the motivations behind this is that it is much easier to solve for forces such as pressure on a uniform grid, but it makes more sense to keep track of specific particle attributes such as position and velocity on the particles themselves. Thus, for every frame, we project our particle attributes onto the grid, compute forces and presssure on the grid, and then interpolate updated velocities back onto the particles before advection. The PIC/FLIP part comes in how we update with new velocities. With the PIC approach, we simply update particles with the new grid velocity. With the FLIP approach, we increment the particle velocity by the delta in grid velocity. The latter better maintains individual particle velocities but can become chaotic, so it is common to use a blend of 5% PIC and 95% FLIP.</p> <p dir="auto">The following will be an abbreviated breakdown of how all of the steps in the fluid solve are accomplished in shaders.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Particle-to-Grid</h3><a id="user-content-particle-to-grid" class="anchor" aria-label="Permalink: Particle-to-Grid" href="#particle-to-grid"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">This first step is a fairly tricky and annoying part of the solve. For every particle, we need to project it's velocity to possibly 3x27 = 81 different locations. There are three grids to project onto (U, V, W for X, Y, Z velocities) but we need to project onto the 27 cells in our direct neighborhood. To do this, we do exactly that. For every particle, we render 81 gl.POINTS. In practice, this is calling <code>gl.drawArrays(pointBuffer, 0, 81*pointCount)</code>.</p> <p dir="auto">We number these points with increasing values so that in the vertex shader we can correctly compute which grid we are to project onto. After calculating the correct grid and grid location, we compute the UV coordinate of that position and set glPosition accordingly.</p> <p dir="auto">Note that we want each grid cell to store the weighted average of all the nearby particles so we actually need to accumulate all of the nearby velocities and keep track of their weights. To do this, we preface our draw call with <code>gl.enable(gl.BLEND)</code> and <code>gl.blendFunc(gl.ONE, gl.ONE)</code>. This will allow us to take advantage of hardware blending functions to accumulate influences. In practice, we do this draw call twice, writing to different framebuffers. To one we write velocity * weight, and to the other we write just the weight. Finally, we bind both textures and render a texture-sized quad, dividing all accumulated velocities by their respective cumulative weight.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Copy-Grid</h3><a id="user-content-copy-grid" class="anchor" aria-label="Permalink: Copy-Grid" href="#copy-grid"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">At this point, we copy our velocity field to a new texture. This is important for later computing the change in grid velocity.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Mark-Cells</h3><a id="user-content-mark-cells" class="anchor" aria-label="Permalink: Mark-Cells" href="#mark-cells"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">In this step we mark cells with the type of their contents: either SOLID, FLUID, or AIR. Marking cells is important because we use this information later to set up our pressure equations. To accomplish this, we do a similar scatter-approach taken in the particle-to-grid transfer. All cells are originally air (value is 0). We then render 7 points for each particle. We use 7 because we actually want to mark the cells at the fluid boundary as fluid as well, we we extend the fluid-marking in each cardinal direction. Using the point index, we compute the grid cell we want to project onto and write a value of 1.0 (FLUID) to the computed UV coordinate. If there are solids in the scene, we do the same thing for them. Note: we <strong>do not</strong> use <code>gl.blend(gl.ONE, gl.ONE)</code>. We do not want to accumulate the values of cell types.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Gravity-Update</h3><a id="user-content-gravity-update" class="anchor" aria-label="Permalink: Gravity-Update" href="#gravity-update"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">This is a fairly simple step where we just render a fullscreen quad and increment the y-velocities by G * timestep. In practice, I only update velocities near non-air cells or the debug views become difficult to view.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Pressure-solve</h3><a id="user-content-pressure-solve" class="anchor" aria-label="Permalink: Pressure-solve" href="#pressure-solve"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">This is the most computational intensive part of the fluid solve. This is fairly complicated (it is definitely advised to read Bridson's book), but in short, we want to solve a pressure equation that will result in 0 divergence within the fluid. In other words, we want to compute pressures that will cause the fluid to move in such a way that it does not lose or gain any volume.</p> <p dir="auto">This comes down to solving a system <code>Ap = b</code>, where <code>b</code> is a vector of divergences (total change in velocity across a cell), and <code>A</code> is a coefficient matrix. If we have an <code>X x Y x Z</code> grid, the dimensions of <code>A</code> are <code>X*Y*Z x X*Y*Z</code>. If you work through the pressure equations, we realize that the diagonal entries of A denote the number of non-solid neighbors that cell <code>(i, j, k)</code> has, and the entries of the form <code><(i1, j1, k1), (i2, j2, k2)></code> are <code>-1</code> if those two cells are both fluid and <code>0</code> otherwise. Fortunately, this means that <code>A</code> is a very, very sparse matrix because each cell has at most 6 neighbors. Every row and column in <code>A</code> has <strong>at most</strong> 7 entries. Furthermore, A is symmetric so half of the non-diagonal entries don't need to be stored. This ultimately comes down to just 4 entries per cell, which is easily stored in the RGBA components of a single texture!</p> <p dir="auto">The system is solved using the <a href="https://en.wikipedia.org/wiki/Conjugate_gradient_method#The_preconditioned_conjugate_gradient_method" rel="nofollow">preconditioned conjugate gradient method</a> which is an iterative method for solving systems like this (symmetric, positive-definite). The details are better explained elsewhere but I will briefly explain how some of the important steps are accomplished.</p> <p dir="auto"><strong>Preconditioning</strong></p> <p dir="auto">Preconditioning is an extremely important step of the solve which makes it signficiantly faster to solve. <code>A</code> is typically not an invertible matrix, but preconditioning constructs an approximate inverse of <code>A</code> which enables us to get much closer to the correct solution in less time. Typically, CPU solvers use an Incomplete Cholesky preconditioner which works very, very well. However, it is not well suited to a GPU or shader implementation because it requires doing two triangular matrix solves with forward and backsubstitution for every iteration. On a GPU this is absolutely terrible and would involve hundreds of serial kernel invocations or draw calls. Instead, we use the Incomplete Poisson preconditioner which is accomplished with just two draw calls.</p> <p dir="auto">Given a residual vector <code>r</code>, we want compute <code>z = Mr</code>, where <code>M</code> is an approximate inverse of <code>A</code>. The Incomplete Poisson preconditioner says that M = (I - LD<sup>-1</sup>)(I - D<sup>-1</sup>L) where <code>L</code> i strictly the lower diagonal entries of <code>A</code> and <code>D</code> is strictly the diagonal entires of <code>A</code>. We just need to compute z = (I - LD<sup>-1</sup>)(I - D<sup>-1</sup>L)r which is just two draw calls, one for each matrix multiply.</p> <p dir="auto">For each entry in <code>r</code>:</p> <ol dir="auto"> <li>Set <code>z[i] = r[i] - r * L[i, :] * 1/d</code>. This is easy because <code>L</code> is sparse and has just 3 items in row i. <code>d</code> is just the diagonal entry corresponding to <em>row</em> i.</li> <li>Set <code>z[i] = z[i] - z * L[i, :] * 1/d</code>. This is easy because <code>L</code> is sparse and has just 3 items in row i. <code>d</code> is just the diagonal entry corresponding to <em>column</em> i.</li> </ol> <p dir="auto"><strong>Dot Products</strong></p> <p dir="auto">Unfortunately, we need to take three dot products every iteration, which is not a very parallel operation. We once again take advantage of texture blending and using <code>gl.blend(gl.ONE, gl.ONE)</code> and write all products to the same UV coordinate to accumulate them.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Velocity Extrapolation</h3><a id="user-content-velocity-extrapolation" class="anchor" aria-label="Permalink: Velocity Extrapolation" href="#velocity-extrapolation"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">In this step we extrapolate velocities to push the velocities one grid cell past the fluid cells. This is important so that particles don't lose energy when they come in contact with air cells. This is implemented in a fragment shader as a gather operation which does a few grid type checks in the cardinal directions and appropriately updates velocity.</p> <div class="markdown-heading" dir="auto"><h3 tabindex="-1" class="heading-element" dir="auto">Grid-To-Particle</h3><a id="user-content-grid-to-particle" class="anchor" aria-label="Permalink: Grid-To-Particle" href="#grid-to-particle"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto">The grid-to-particle transfer is much simpler than the particle-to-grid transfer. For this, we only need to compute our fractional grid cell index and then trilinear-ly interpolate our new updated velocity.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Performance</h2><a id="user-content-performance" class="anchor" aria-label="Permalink: Performance" href="#performance"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <p dir="auto"><strong>Tested on: Windows 10, i7-4770K @ 3.50GHz 16GB, GTX 780 3072MB</strong></p> <markdown-accessiblity-table><table> <thead> <tr> <th>Grid Density</th> <th>FPS (Incomplete Cholesky)</th> <th>FPS (Incomplete Poisson)</th> </tr> </thead> <tbody> <tr> <td>5,000</td> <td>40</td> <td>60</td> </tr> <tr> <td>10,000</td> <td>30</td> <td>60</td> </tr> <tr> <td>50,000</td> <td>9</td> <td>33</td> </tr> <tr> <td>100,000</td> <td>3</td> <td>17</td> </tr> <tr> <td>500,000</td> <td>0.5</td> <td>6</td> </tr> <tr> <td>1,000,000</td> <td>0.2</td> <td>3</td> </tr> </tbody> </table></markdown-accessiblity-table> <p dir="auto">As we can see, using the Incomplete Poisson preconditioner has signficant performance benefits and is able to more effectively handle high grid resolutions.</p> <p dir="auto">Keeping grid density constant but doubling particle density, these numbers are exactly the same, confirming that the limiting factor for performance is the pressure solve, which is bound by grid density.</p> <div class="markdown-heading" dir="auto"><h2 tabindex="-1" class="heading-element" dir="auto">Build</h2><a id="user-content-build" class="anchor" aria-label="Permalink: Build" href="#build"><svg class="octicon octicon-link" viewBox="0 0 16 16" version="1.1" width="16" height="16" aria-hidden="true"><path d="m7.775 3.275 1.25-1.25a3.5 3.5 0 1 1 4.95 4.95l-2.5 2.5a3.5 3.5 0 0 1-4.95 0 .751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018 1.998 1.998 0 0 0 2.83 0l2.5-2.5a2.002 2.002 0 0 0-2.83-2.83l-1.25 1.25a.751.751 0 0 1-1.042-.018.751.751 0 0 1-.018-1.042Zm-4.69 9.64a1.998 1.998 0 0 0 2.83 0l1.25-1.25a.751.751 0 0 1 1.042.018.751.751 0 0 1 .018 1.042l-1.25 1.25a3.5 3.5 0 1 1-4.95-4.95l2.5-2.5a3.5 3.5 0 0 1 4.95 0 .751.751 0 0 1-.018 1.042.751.751 0 0 1-1.042.018 1.998 1.998 0 0 0-2.83 0l-2.5 2.5a1.998 1.998 0 0 0 0 2.83Z"></path></svg></a></div> <ol dir="auto"> <li><a href="https://nodejs.org/en/download/" rel="nofollow">Install <code>Node.js</code></a></li> <li>Fork and clone this repository.</li> <li><code>npm install</code> in the project root directory.</li> <li><code>npm run build</code> and open <code>demo/index.html</code> OR <code>npm start</code> and navigate to <code>localhost:3000/demo</code></li> </ol> </article></div></div></div></div></div> <!-- --> <!-- --> <script type="application/json" id="__PRIMER_DATA_:R0:__">{"resolvedServerColorMode":"day"}</script></div> </react-partial> <input type="hidden" data-csrf="true" value="0mwRWkwafOSrefp01PJgjYytAFF5zpnCjULI1qQ84bhDJbd1nXcFAorSf7+mQdYc3FDy1OeeUvBwrm1QZ+zUqg==" /> </div> <div data-view-component="true" class="Layout-sidebar"> <div class="BorderGrid about-margin" data-pjax> <div class="BorderGrid-row"> <div class="BorderGrid-cell"> <div class="hide-sm hide-md"> <h2 class="mb-3 h4">About</h2> <p class="f4 my-3"> PIC/FLIP Fluid with WebGL </p> <div class="my-3 d-flex flex-items-center"> <svg aria-hidden="true" height="16" viewBox="0 0 16 16" 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