diff --git a/phases/fem-and-beam-kernel/index.json b/phases/fem-and-beam-kernel/index.json
index b29cc85..8823f5a 100644
--- a/phases/fem-and-beam-kernel/index.json
+++ b/phases/fem-and-beam-kernel/index.json
@@ -35,5 +35,6 @@
       "completed_at": "2026-07-31T02:24:47+0900"
     }
   ],
-  "created_at": "2026-07-31T01:37:23+0900"
+  "created_at": "2026-07-31T01:37:23+0900",
+  "completed_at": "2026-07-31T02:24:48+0900"
 }
\ No newline at end of file
diff --git a/phases/index.json b/phases/index.json
index 883bafb..9fcc70f 100644
--- a/phases/index.json
+++ b/phases/index.json
@@ -12,7 +12,8 @@
     },
     {
       "dir": "fem-and-beam-kernel",
-      "status": "pending"
+      "status": "completed",
+      "completed_at": "2026-07-31T02:24:48+0900"
     },
     {
       "dir": "equation-and-linear-solve",
@@ -43,4 +44,4 @@
       "status": "pending"
     }
   ]
-}
+}
\ No newline at end of file